Lobachevskii Journal of Mathematics Vol.8, 2001, 33–165 \begin{matrix}c \\ ◯ \end{matrix}B. Komrakov, Jnr.

B. Komrakov, Jnr. EINSTEIN–MAXWELL EQUATION ON FOUR-DIMENSIONAL HOMOGENEOUS SPACES

(submitted by M.A.Malakhaltsev)

Abstract. This paper presents the solutions of the Einstein–Maxwell equation on all local four-dimensional pseudo-Riemannian homogeneous spaces and the complete local classification of four-dimensional Einstein–Maxwell homogeneous spaces with an invariant pseudo-Riemannian metric of arbitrary signature.

Table of Contents

Introduction

Chapter I. Pseudo-Riemannian pairs

1. Classification of subalgebras in the Lie algebras 𝔰𝔬(4), 𝔰𝔬(3, 1) and 𝔰𝔬(2, 2)

2. Classification of pseudo-Riemannian pairs

Chapter II. Einstein–Maxwell equation on four-dimensional homogeneous spaces.

Preliminaries

1. Solutions of Einstein–Maxwell equations on pseudo-Riemannian pairs of codimension 4

2. Four-dimensional Einstein–Maxwell homogeneous spaces

References

Introduction

We consider classification of low-dimensional homogeneous spaces as an immediate continuation of classifications obtained by Sophus Lie. Twodimensional homogeneous spaces were classified locally by Sophus Lie [L1] and globally by G.D. Mostow [M]. (See also preprint [KTD], where the complete classification of two-dimensional homogeneous spaces, both locally and globally, is presented.) S. Lie also obtained some results in classification of three-dimensional homogeneous spaces and described all subalgebras in the Lie algebra 𝔰𝔬(4, ℂ) (in terms of vector fields). A detailed account of these classifications can be found in [L2]. The local classification of all threedimensional isotropically-faithful homogeneous spaces was obtained in [KT], and the classification (local and global) of all two- and three-dimensional pseudo-Riemannian isotropically-faithful homogeneous spaces was given in [DK].

The problem of classification of four-dimensional pseudo-Riemannian homogeneous spaces is interesting from the point of view of both geometry and physics, and not only in the case of signature (1, 3) (spaces of relativity theory) but also in the case of signature (2, 2) (twistors).

The classification (local and global) of all four-dimensional Riemannian homogeneous spaces can be found in [B, I], and the local classification of all complex four-dimensional homogeneous spaces with an invariant symmetric bilinear form was obtained in [K1] (for a summary of results see [K]). The complete local classification of four-dimensional homogeneous spaces with an invariant pseudo-Riemannian metric of arbitrary signature was presented in [K2]. We recall this classification in Chapter I.

Let (\overline{G}, M) be a homogeneous space, G = \overline{G}_{x} the stabilizer of an arbitrary point x ∈ M, and (𝔤̄, 𝔤) the pair of Lie algebras corresponding to the pair (\overline{G}, G) of Lie groups. Note that the pair (𝔤̄, 𝔤) locally uniquely defines the homogeneous space (\overline{G}, M). A global description of all homogeneous spaces corresponding to the given pair can be found in [M].

The Einstein–Maxwell equation on the homogeneous space (\overline{G}, M) has the form:

r − λg = m_{ω},  (1)

where g is an invariant pseudo-Riemannian metric, ω is an invariant differential 2-form on M which is closed (dω = 0) and coclosed (d(∗ω) = 0), λ is an arbitrary real number, r is the Ricci tensor of g, m_{ω} is an invariant smooth tensor field of type (0,2) on M defined by m_{ω} = ω ∘ g^{−1} ∘ ω, where g, ω, m_{ω} are identified with the corresponding mappings T M → T^{∗}M.

Equation (1) is called the Einstein equation if m_{ω} ≡ 0.

The global classification of all four-dimensional Riemannian Einstein homogeneous spaces was given in [J], and the classification of all isotropy irreducible Riemannian homogeneous spaces, which are obviously Einstein spaces, can be found in [W] (See also [Be]).

Let (𝔤̄, 𝔤) be a pair of Lie algebras corresponding to the homogeneous space (\overline{G}, M). By 𝔪 denote the factor space 𝔤̄/𝔤 endowed with the natural structure of 𝔤-module. Then, in terms of the pair (𝔤̄, 𝔤), the Einstein–Maxwell equation on the homogeneous space (\overline{G}, M) will have the form:

R − λB = M_{Ω},

where B is an invariant nondegenerate symmetric bilinear form on 𝔪, Ω is an invariant skew-symmetric bilinear form on 𝔪 which is closed (dΩ = 0) and coclosed (d(∗Ω) = 0), R is the invariant symmetric bilinear form on 𝔪 corresponding to the Ricci tensor, λ is an arbitrary real number, M_{Ω} is the invariant symmetric bilinear form on 𝔪 corresponding to m_{ω}.

The solution of the Einstein–Maxwell equation on the pair (𝔤̄, 𝔤) is any triple (B, Ω, λ) satisfying the equality R − λB = M_{Ω}.

This paper presents the solutions of the Einstein–Maxwell equation on all local four-dimensional pseudo-Riemannian homogeneous spaces. We give this results in Chapter II, 1.

A pseudo-Riemannian homogeneous space (\overline{G}, M) is said to be an Einstein– Maxwell homogeneous space if the Einstein–Maxwell equation has a solution on (\overline{G}, M). We can divide the four-dimensional Einstein–Maxwell homogeneous spaces into the Riemannian (with an invariant Riemannian metric), the Lorentzian (with an invariant pseudo-Riemannian metric of signature (3,1)), and the spaces of type (2,2) (with an invariant pseudo-Riemannian metric of signature (2,2)).

This paper presents the complete local classification of four-dimensional Einstein–Maxwell homogeneous spaces with an invariant pseudo-Riemannian metric of arbitrary signature. We give these classifications in Chapter II, 2.

CHAPTER I PSEUDO-RIEMANNIAN PAIRS

1. Classification of subalgebras in the Lie algebras 𝔰𝔬(4), 𝔰𝔬(3, 1) and 𝔰𝔬(2, 2)

Preliminaries:

1. For the sake of simplicity instead of the standard notation for a subalgebra of 𝔰𝔬(2, 2) (𝔰𝔬(4) or 𝔰𝔬(3, 1)) such as

𝔤 = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \end{matrix} ) | x ∈ ℝ, λ ∈ [0, 1]\}

we use the following notation:

𝔤 = ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \end{matrix} ) , λ ∈ [0, 1].

Here we imply that variables denoted by Latin letters run through ℝ and that parameters are denoted by small Greek letters.

2. To refer to subalgebras determined in Theorem 1 we use the following notation:

d.n^{k},

where d is the dimension of the subalgebra; n is the number of the complex subalgebra 𝔤^{ℂ} of 𝔰𝔬(4, ℂ); k is the number of the real form of 𝔤^{ℂ} in Theorem 1.

Theorem 1. Any nonzero subalgebra of the Lie algebra 𝔰𝔬(4) is conjugate (with respect to GL(4, ℝ)) to one and only one of the following subalgebras:

\underline{dim 𝔤 = 1}

1.1^{2} ( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \\ x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \end{matrix} ) , λ ∈ [0, 1]

\underline{dim 𝔤 = 2}

2.1^{3} ( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & −y \\ x & 0 & 0 & 0 \\ 0 & y & 0 & 0 \end{matrix} )

\underline{dim 𝔤 = 3}

3.4^{2} ( \begin{matrix}0 & y & −x & −z \\ −y & 0 & −z & x \\ x & z & 0 & y \\ z & −x & −y & 0 \end{matrix} ) 3.5^{2} ( \begin{matrix}0 & x & y & 0 \\ −x & 0 & z & 0 \\ −y & −z & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

\underline{dim 𝔤 = 4}

4.2^{2} ( \begin{matrix}0 & y & −x & −z \\ −y & 0 & −z & −t \\ x & z & 0 & y \\ z & t & −y & 0 \end{matrix} )

\underline{dim 𝔤 = 6}

6.1^{2} ( \begin{matrix}0 & x & y & z \\ −x & 0 & t & u \\ −y & −t & 0 & v \\ −z & −u & −v & 0 \end{matrix} )

Any nonzero subalgebra of the Lie algebra 𝔰𝔬(3, 1) is conjugate (with respect to GL(4, ℝ)) to one and only one of the following subalgebras:

\underline{dim 𝔤 = 1}

1.1^{1} (λ = 0) ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) 1.1^{2} (λ = 0) ( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \\ x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

1.1^{3} ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & 0 & −λx \\ 0 & 0 & −x & 0 \\ 0 & λx & 0 & 0 \end{matrix} ) 1.1^{4} ( \begin{matrix}0 & 0 & −x & 0 \\ 0 & −λx & 0 & 0 \\ x & 0 & 0 & 0 \\ 0 & 0 & 0 & λx \end{matrix} )

λ ∈]0, 1] λ ∈]0, 1[

1.4^{1} ( \begin{matrix}0 & x & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

\underline{dim 𝔤 = 2}

2.1^{2} ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & 0 & −y \\ 0 & 0 & −x & 0 \\ 0 & y & 0 & 0 \end{matrix} ) 2.4^{1} ( \begin{matrix}x & y & 0 & 0 \\ 0 & 0 & y & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

2.5^{2} ( \begin{matrix}0 & x & 0 & −y \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & y & 0 \end{matrix} )

\underline{dim 𝔤 = 3}

3.2^{2} ( \begin{matrix}x & y & 0 & −z \\ 0 & 0 & −y & −λx \\ 0 & 0 & −x & 0 \\ 0 & λx & z & 0 \end{matrix} ) 3.3^{2} ( \begin{matrix}0 & y & 0 & −z \\ 0 & 0 & −y & −x \\ 0 & 0 & 0 & 0 \\ 0 & x & z & 0 \end{matrix} )

λ ⩾ 0

3.5^{1} ( \begin{matrix}2x & y & 0 & 0 \\ 2z & 0 & −2y & 0 \\ 0 & −z & −2x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) 3.5^{2} ( \begin{matrix}0 & x & y & 0 \\ −x & 0 & z & 0 \\ −y & −z & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

4.1^{2} ( \begin{matrix}x & z & 0 & −t \\ 0 & 0 & −z & −y \\ 0 & 0 & −x & 0 \\ 0 & y & t & 0 \end{matrix} )

\underline{dim 𝔤 = 6}

6.1^{3} ( \begin{matrix}0 & x & y & z \\ −x & 0 & t & u \\ −y & −t & 0 & v \\ z & u & v & 0 \end{matrix} )

Any nonzero subalgebra of the Lie algebra 𝔰𝔬(2, 2) is conjugate (with respect to GL(4, ℝ)) to one and only one of the following subalgebras:

\underline{dim 𝔤 = 1}

1.1^{1} ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \end{matrix} ) , λ ∈ [0, 1]  1.1^{2} ( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \\ x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \end{matrix} ) , λ ∈ [0, 1]

1.2^{1} ( \begin{matrix}x & x & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −x & −x \end{matrix} )  1.2^{2} ( \begin{matrix}0 & −x & 0 & −x \\ x & 0 & x & 0 \\ 0 & 0 & 0 & −x \\ 0 & 0 & x & 0 \end{matrix} )

1.3^{1} ( \begin{matrix}0 & 0 & x & 0 \\ 0 & 0 & 0 & x \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )  1.4^{1} ( \begin{matrix}0 & x & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

1.1^{5} ( \begin{matrix}x cos \frac{ϕ}{2} & x sin \frac{ϕ}{2} & 0 & 0 \\ −x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} & 0 & 0 \\ 0 & 0 & −x cos \frac{ϕ}{2} & −x sin \frac{ϕ}{2} \\ 0 & 0 & x sin \frac{ϕ}{2} & −x cos \frac{ϕ}{2} \end{matrix} ) , ϕ ∈]0, \frac{π}{2} ]

1.1^{6} ( \begin{matrix}−x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} & 0 & 0 \\ −x cos \frac{ϕ}{2} & −x sin \frac{ϕ}{2} & 0 & 0 \\ 0 & 0 & x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} \\ 0 & 0 & −x cos \frac{ϕ}{2} & x sin \frac{ϕ}{2} \end{matrix} ) , ϕ ∈]0, \frac{π}{2} [

2.1^{1} ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & y & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & −y \end{matrix} )  2.1^{3} ( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & −y \\ x & 0 & 0 & 0 \\ 0 & y & 0 & 0 \end{matrix} )

2.1^{4} ( \begin{matrix}x & −y & 0 & 0 \\ y & x & 0 & 0 \\ 0 & 0 & −x & y \\ 0 & 0 & −y & −x \end{matrix} )  2.2^{1} ( \begin{matrix}x & y & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & −λx \end{matrix} ) , λ ∈ [−1, 1]

2.2^{2} ( \begin{matrix}0 & −x & y & 0 \\ x & 0 & 0 & y \\ 0 & 0 & 0 & −x \\ 0 & 0 & x & 0 \end{matrix} )  2.3^{1} ( \begin{matrix}x & y & 0 & x \\ 0 & −x & −x & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & x \end{matrix} )

2.4^{1} ( \begin{matrix}x & y & 0 & 0 \\ 0 & 0 & y & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )  2.5^{1} ( \begin{matrix}0 & x & 0 & y \\ 0 & 0 & −y & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 \end{matrix} )

2.2^{3} ( \begin{matrix}−x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} & y & 0 \\ −x cos \frac{ϕ}{2} & −x sin \frac{ϕ}{2} & 0 & y \\ 0 & 0 & x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} \\ 0 & 0 & −x cos \frac{ϕ}{2} & x sin \frac{ϕ}{2} \end{matrix} ) , ϕ ∈]0, π[

\underline{dim 𝔤 = 3}

3.1^{1} ( \begin{matrix}x & z & 0 & 0 \\ 0 & y & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −z & −y \end{matrix} )  3.1^{2} ( \begin{matrix}x & −y & z & 0 \\ y & x & 0 & z \\ 0 & 0 & −x & −y \\ 0 & 0 & y & −x \end{matrix} )

3.2^{1} ( \begin{matrix}x & y & 0 & z \\ 0 & λx & −z & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & −λx \end{matrix} ) , λ ⩾ 0  3.3^{1} ( \begin{matrix}0 & y & 0 & z \\ 0 & x & −z & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −y & −x \end{matrix} )

3.4^{1} ( \begin{matrix}x & y & 0 & 0 \\ z & −x & 0 & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & x \end{matrix} )  3.5^{1} ( \begin{matrix}2x & y & 0 & 0 \\ 2z & 0 & −2y & 0 \\ 0 & −z & −2x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

\underline{dim 𝔤 = 4}

4.1^{1} ( \begin{matrix}x & z & 0 & t \\ 0 & y & −t & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −z & −y \end{matrix} ) 4.2^{1} ( \begin{matrix}x & y & 0 & 0 \\ z & t & 0 & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & −t \end{matrix} )

4.3^{1} ( \begin{matrix}x & y & 0 & t \\ z & −x & −t & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & x \end{matrix} )

\underline{dim 𝔤 = 5}

5.1^{1} ( \begin{matrix}x & y & 0 & u \\ z & t & −u & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & −t \end{matrix} )

\underline{dim 𝔤 = 6}

6.1^{1} ( \begin{matrix}x & y & 0 & u \\ z & t & −u & 0 \\ 0 & v & −x & −z \\ −v & 0 & −y & −t \end{matrix} )

2. Classification of pseudo-Riemannian pairs

Preliminaries:

1. Let 𝔤 be one of the subalgebras of the Lie algebras 𝔰𝔬(4), 𝔰𝔬(3, 1) or 𝔰𝔬(2, 2) (𝔰𝔬(p, q), p + q = 4) determined in Theorem 1. We assume that the Lie algebra 𝔤 acts naturally on ℝ^{4} ; then (𝔤, ℝ^{4}) is a faithful generalized module. The enumeration of the generalized modules obtained in this way coincide with that of the corresponding subalgebras of 𝔰𝔬(p, q) in Theorem 1.

We say that a pair (𝔤̄, 𝔤) has type n.m^{k}, if the corresponding generalized module (𝔤, 𝔤̄/𝔤) is isomorphic to the generalized module n.m^{k}, i.e., to the generalized module (𝔤, ℝ^{4}), where 𝔤 is the subalgebra of 𝔰𝔬(p, q) supplied with the number n.m^{k} in Theorem 1.

2. Let (𝔤̄, 𝔤) be a pair of type n.m^{k}. Then without loss of generality we can identify the Lie algebra 𝔤 with the subalgebra n.m^{k} of 𝔰𝔬(p, q).

Let {u_{1}, u_{2}, u_{3}, u_{4}} be the standard basis of ℝ^{4}:

u_{1} = ( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , u_{2} = ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , u_{3} = ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , u_{4} = ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) .

3. We define a pair (𝔤̄, 𝔤) in two ways: a) By the commutation table of the Lie algebra 𝔤̄ only. Here by {e_{1}, …, e_{n}, u_{1}, u_{2}, u_{3}, u_{4}} we denote a basis of 𝔤̄ (n = dim 𝔤). We assume that the Lie algebra 𝔤 is generated by e_{1}, …, e_{n}.

b) A more compact and generally accepted way, using the decomposition of the Lie algebra 𝔤̄ in direct (semidirect) product of more simple Lie algebras.

By p, r, s, etc. we denote the parameters appearing in the process of the classification. If there are some complementary conditions on them, it is indicated just after the table. Otherwise we assume that these parameters run through ℝ.

4. By 𝔯_{2} = ⟨p, q⟩ we denote the following Lie algebra: [p, q] = p. By 𝔫_{3} = ⟨h, p, q⟩ we denote the following Lie algebra:

\begin{matrix}[ ,] & h & p & q \\ h & 0 & 0 & 0 \\ p & 0 & 0 & h \\ q & 0 & −h & 0 \end{matrix} .

By 𝔫_{4} = ⟨h, p_{1}, p_{2}, p_{3}⟩ we denote the following Lie algebra:

\begin{matrix}[ ,] & h & p_{1} & p_{2} & p_{3} \\ h & 0 & 0 & p_{1} & p_{2} \\ p_{1} & 0 & 0 & 0 & 0 \\ p_{2} & −p_{1} & 0 & 0 & 0 \\ p_{3} & −p_{2} & 0 & 0 & 0 \end{matrix} .

By 𝔫_{5} = ⟨h, p_{1}, p_{2}, q_{1}, q_{2}⟩ we denote the following Lie algebra:

\begin{matrix}[ ,] & h & p_{1} & p_{2} & q_{1} & q_{2} \\ h & 0 & 0 & 0 & 0 & 0 \\ p_{1} & 0 & 0 & 0 & h & 0 \\ p_{2} & 0 & 0 & 0 & 0 & h \\ q_{1} & 0 & −h & 0 & 0 & 0 \\ q_{2} & 0 & 0 & −h & 0 & 0 \end{matrix} .

By 𝔫̃_{5} = ⟨h, p_{1}, p_{2}, q_{1}, q_{2}⟩ we denote the following Lie algebra:

\begin{matrix}[ ,] & h & p_{1} & p_{2} & q_{1} & q_{2} \\ h & 0 & 0 & 0 & p_{1} & p_{2} \\ p_{1} & 0 & 0 & 0 & 0 & 0 \\ p_{2} & 0 & 0 & 0 & 0 & 0 \\ q_{1} & −p_{1} & 0 & 0 & 0 & h \\ q_{2} & −p_{2} & 0 & 0 & −h & 0 \end{matrix} .

5. We consider only the case 𝔤 ≠ 0. Theorem 2. Any pseudo-Riemannian pair (𝔤̄, 𝔤) of codimension 4 is equivalent to one and only one of the following pairs: 1.1^{1}

𝔤 = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \end{matrix} )| x ∈ ℝ, λ ∈ [0, 1]\}

λ = 0

1.

𝔤̄ = \{( \begin{matrix}y & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & y − x \end{matrix} )| x, y ∈ ℝ\}⋌𝔫_{3}, 𝔤 = \{( \begin{matrix}0 & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & −x \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{3} & −u_{2} & 0 & 0 & u_{3} \\ u_{4} & 0 & 0 & −u_{2} & −u_{3} & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 \\ 0 & y − x & 0 \\ 0 & 0 & py \end{matrix} )| x, y ∈ ℝ, p ∈ ℝ\}⋌ℝ^{3}, 𝔤 = \{( \begin{matrix}x & 0 & 0 \\ 0 & −x & 0 \\ 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & pu_{2} \\ u_{3} & u_{3} & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & 0 & −pu_{2} & −u_{3} & 0 \end{matrix}

3.

𝔤̄ = 𝔰𝔩(2, ℝ) × ℝ × ℝ, 𝔤 = \{[( \begin{matrix}x & 0 \\ 0 & −x \end{matrix} ) , x, 0]| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & e_{1}+u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & −e_{1}−u_{2} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

4.

𝔤̄ = (𝔤 ⋌ 𝔫_{3}) × ℝ, 𝔤 = \{( \begin{matrix}0 & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & −x \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & −u_{2} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

5.

𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔯_{2}, 𝔤 = 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{3} & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{2} & 0 & 0 \end{matrix}

6. 𝔤̄ = (𝔰𝔬(1, 1) ⋌ ℝ^{2}) × 𝔯_{2}, 𝔤 = 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{2} & 0 & 0 \end{matrix}

7. 𝔤̄ = 𝔰𝔩(2, ℝ) × ℝ^{2}, 𝔤 = 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

λ = \frac{1}{2}

8.

𝔤̄ = 𝔰𝔩(2, ℝ) ⋌ ℝ^{2}, 𝔤 = 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & \frac{1}{2}u_{2} & −u_{3} & −\frac{1}{2}u_{4} \\ u_{1} & −u_{1} & 0 & 0 & −2e_{1} & u_{2} \\ u_{2} & −\frac{1}{2}u_{2} & 0 & 0 & u_{4} & 0 \\ u_{3} & u_{3} & 2e_{1} & −u_{4} & 0 & 0 \\ u_{4} & \frac{1}{2}u_{4} & −u_{2} & 0 & 0 & 0 \end{matrix}

9.

𝔤̄ = 𝔤 ⋌ (𝔫_{3} × ℝ), 𝔤 = \{( \begin{matrix}\frac{x}{2} & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & −\frac{x}{2} & 0 \\ 0 & 0 & 0 & −x \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & \frac{1}{2}u_{2} & −u_{3} & −\frac{1}{2}u_{4} \\ u_{1} & −u_{1} & 0 & 0 & 0 & u_{2} \\ u_{2} & −\frac{1}{2}u_{2} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 \\ u_{4} & \frac{1}{2}u_{4} & −u_{2} & 0 & 0 & 0 \end{matrix}

λ ∈ [0, 1]

10. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.1^{1}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & λu_{2} & −u_{3} & −λu_{4} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{2} & −λu_{2} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 \\ u_{4} & λu_{4} & 0 & 0 & 0 & 0 \end{matrix}

1.1^{2}

𝔤 = \{( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & −λx \\ x & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \end{matrix} )| x ∈ ℝ, λ ∈ [0, 1]\}

46 B. KOMRAKOV, JNR.

λ = 0

1.

𝔤̄ = \{( \begin{matrix}2y & 0 & 0 \\ 0 & y & x \\ 0 & −x & y \end{matrix} )| x, y ∈ ℝ\}⋌𝔫_{3}, 𝔤 = \{( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & x \\ 0 & −x & 0 \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & −u_{2} & u_{1} \\ u_{2} & 0 & 0 & 0 & 0 & 2u_{2} \\ u_{3} & u_{1} & u_{2} & 0 & 0 & u_{3} \\ u_{4} & 0 & −u_{1} & −2u_{2} & −u_{3} & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}y & x & 0 \\ −x & y & 0 \\ 0 & 0 & py \end{matrix} )| x, y ∈ ℝ, p ∈ ℝ\}⋌ℝ^{3}, 𝔤 = \{( \begin{matrix}0 & x & 0 \\ −x & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & 0 & pu_{2} \\ u_{3} & u_{1} & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & −u_{1} & −pu_{2} & −u_{3} & 0 \end{matrix}

3.

𝔤̄ = 𝔰𝔲(2) × ℝ × ℝ, 𝔤 = \{[( \begin{matrix}0 & x \\ −x & 0 \end{matrix} ) , x, 0]| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & e_{1} + u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & −e_{1} − u_{2} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

4.

𝔤̄ = 𝔰𝔩(2, ℝ) × ℝ × ℝ, 𝔤 = \{[( \begin{matrix}0 & x \\ −x & 0 \end{matrix} ) , x, 0]| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & −e_{1} + u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & e_{1} − u_{2} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

5.

𝔤̄ = (𝔤 ⋌ 𝔫_{3}) × ℝ, 𝔤 = \{( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & −x \\ 0 & x & 0 \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & −u_{2} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

6.

𝔤̄ = 𝔰𝔲(2) × 𝔯_{2}, 𝔤 = 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{1} & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{2} & 0 & 0 \end{matrix}

7.

𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔯_{2}, 𝔤 = 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & −e_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{1} & e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{2} & 0 & 0 \end{matrix}

8.

𝔤̄ = (𝔰𝔬(2) ⋌ ℝ^{2}) × 𝔯_{2}, 𝔤 = 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{2} & 0 & 0 \end{matrix}

9.

𝔤̄ = 𝔰𝔲(2) × ℝ^{2}, 𝔤 = 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

10. 𝔤̄ = 𝔰𝔩(2, ℝ) × ℝ^{2}, 𝔤 = 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & 0 & −u_{1} & 0 \\ u_{1} & −u_{3} & 0 & 0 & −e_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

λ = \frac{1}{2} 11.

𝔤̄ = 𝔰𝔩(2, ℝ) ⋌ ℝ^{2}, 𝔤 = 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & \frac{1}{2}u_{4} & −u_{1} & −\frac{1}{2}u_{2} \\ u_{1} & −u_{3} & 0 & u_{2} & −4e_{1} & −u_{4} \\ u_{2} & −\frac{1}{2}u_{4} & −u_{2} & 0 & −u_{4} & 0 \\ u_{3} & u_{1} & 4e_{1} & u_{4} & 0 & u_{2} \\ u_{4} & \frac{1}{2}u_{2} & u_{4} & 0 & −u_{2} & 0 \end{matrix}

λ ∈ [0, 1]

12. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.1^{2}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & λu_{4} & −u_{1} & −λu_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & 0 \\ u_{2} & −λu_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & λu_{2} & 0 & 0 & 0 & 0 \end{matrix}

1.1^{3}

𝔤 = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & 0 & −λx \\ 0 & 0 & −x & 0 \\ 0 & λx & 0 & 0 \end{matrix} )| x ∈ ℝ, λ ∈]0, 1]\}

1.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.1^{3}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & λu_{4} & −u_{3} & −λu_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{2} & −λu_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 \\ u_{4} & λu_{2} & 0 & 0 & 0 & 0 \end{matrix}

50 B. KOMRAKOV, JNR.

1.1^{4}

𝔤 = \{( \begin{matrix}0 & 0 & −x & 0 \\ 0 & −λx & 0 & 0 \\ x & 0 & 0 & 0 \\ 0 & 0 & 0 & λx \end{matrix} )| x ∈ ℝ, λ ∈]0, 1[\}

1.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.1^{4}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{3} & −λu_{2} & −u_{1} & λu_{4} \\ u_{1} & −u_{3} & 0 & 0 & 0 & 0 \\ u_{2} & λu_{2} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & −λu_{4} & 0 & 0 & 0 & 0 \end{matrix}

1.1^{5}

𝔤 = \{( \begin{matrix}x cos \frac{ϕ}{2} & x sin \frac{ϕ}{2} & 0 & 0 \\ −x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} & 0 & 0 \\ 0 & 0 & −x cos \frac{ϕ}{2} & −x sin \frac{ϕ}{2} \\ 0 & 0 & x sin \frac{ϕ}{2} & −x cos \frac{ϕ}{2} \end{matrix} )| x ∈ ℝ, ϕ ∈]0, \frac{π}{2} ]\}

1.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.1^{5}
[ ,]  e_{1}  u_{1}  u_{2}  u_{3}  u_{4}
e_{1}  0  cos \frac{ϕ}{2}u_{1}−sin \frac{ϕ}{2}u_{2} sin \frac{ϕ}{2}u_{1}+cos \frac{ϕ}{2}u_{2} −cos \frac{ϕ}{2}u_{3}+sin \frac{ϕ}{2}u_{4} −sin \frac{ϕ}{2}u_{3}−cos \frac{ϕ}{2}u_{4}
u_{1} −cos \frac{ϕ}{2}u_{1}+sin \frac{ϕ}{2}u_{2}  0  0  0  0
u_{2} −sin \frac{ϕ}{2}u_{1}−cos \frac{ϕ}{2}u_{2}  0  0  0  0
u_{3}  cos \frac{ϕ}{2}u_{3}−sin \frac{ϕ}{2}u_{4}  0  0  0  0
u_{4}  sin \frac{ϕ}{2}u_{3}+cos \frac{ϕ}{2}u_{4}  0  0  0  0

1.1^{6}

𝔤 = \{( \begin{matrix}−x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} & 0 & 0 \\ −x cos \frac{ϕ}{2} & −x sin \frac{ϕ}{2} & 0 & 0 \\ 0 & 0 & x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} \\ 0 & 0 & −x cos \frac{ϕ}{2} & x sin \frac{ϕ}{2} \end{matrix} )| x ∈ ℝ, ϕ ∈]0, \frac{π}{2} [\}

1.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.1^{6}

EINSTEIN–MAXWELL EQUATION 51

[ ,]  e_{1}  u_{1}  u_{2}  u_{3}  u_{4}
e_{1}  0  −sin \frac{ϕ}{2}u_{1}−cos \frac{ϕ}{2}u_{2} cos \frac{ϕ}{2}u_{1}−sin \frac{ϕ}{2}u_{2} sin \frac{ϕ}{2}u_{3}−cos \frac{ϕ}{2}u_{4} cos \frac{ϕ}{2}u_{3}+sin \frac{ϕ}{2}u_{4}
u_{1}  sin \frac{ϕ}{2}u_{1}+cos \frac{ϕ}{2}u_{2}  0  0  0  0
u_{2} −cos \frac{ϕ}{2}u_{1}+sin \frac{ϕ}{2}u_{2}  0  0  0  0
u_{3} −sin \frac{ϕ}{2}u_{3}+cos \frac{ϕ}{2}u_{4}  0  0  0  0
u_{4} −cos \frac{ϕ}{2}u_{3}−sin \frac{ϕ}{2}u_{4}  0  0  0  0

1.2^{1}

𝔤 = \{( \begin{matrix}x & x & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −x & −x \end{matrix} )| x ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.2^{1}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & u_{1}+u_{2} & −u_{3}−u_{4} & −u_{4} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1}−u_{2} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3}+u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & u_{4} & 0 & 0 & 0 & 0 \end{matrix}

1.2^{2}

𝔤 = \{( \begin{matrix}0 & −x & 0 & −x \\ x & 0 & x & 0 \\ 0 & 0 & 0 & −x \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.2^{2}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{2} & −u_{1} & u_{2} + u_{4} & −u_{1} − u_{3} \\ u_{1} & −u_{2} & 0 & 0 & 0 & 0 \\ u_{2} & u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & −u_{2} − u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & u_{1} + u_{3} & 0 & 0 & 0 & 0 \end{matrix}

52 B. KOMRAKOV, JNR.

1.3^{1}

𝔤 = \{( \begin{matrix}0 & 0 & x & 0 \\ 0 & 0 & 0 & x \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}

1.

𝔤̄ = 𝔰𝔩(2, ℝ) ⋌ ℝ^{2}, 𝔤 = \{( \begin{matrix}0 & 0 \\ z & 0 \end{matrix} )| z ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{1} & 0 & u_{1} & u_{2} \\ u_{1} & −e_{1} & 0 & −\frac{1}{2}u_{2} & u_{3} & \frac{1}{2}u_{4} \\ u_{2} & 0 & \frac{1}{2}u_{2} & 0 & \frac{1}{2}u_{4} & 0 \\ u_{3} & −u_{1} & −u_{3} & −\frac{1}{2}u_{4} & 0 & 0 \\ u_{4} & −u_{2} & −\frac{1}{2}u_{4} & 0 & 0 & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}y & y & λx \\ 0 & 0 & −λx \\ 0 & x & (λ + 1)x \end{matrix} )| x, y ∈ ℝ, |λ| ⩽ 1\} ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −λe_{1}+(λ+1)u_{1}+λu_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{1} & λe_{1}−(λ+1)u_{1}−λu_{2} & 0 & 0 & 0 \\ u_{4} & −u_{2} & 0 & −u_{2} & 0 & 0 \end{matrix}

|λ| ⩽ 1

3.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & x & x \\ −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}⋌[\{( \begin{matrix}y & 0 & y \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )| y ∈ ℝ\} ⋌ ℝ^{3}],

𝔤 = ⟨( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{1} & −u_{1} & 0 & 0 & e_{1} \\ u_{4} & −u_{2} & 0 & −u_{2} & −e_{1} & 0 \end{matrix}

4.

𝔤̄ = \{( \begin{matrix}y & y & (1 + λ^{2})x \\ 0 & 0 & −(1 + λ^{2})x \\ 0 & x & 2λx \end{matrix} )| x, y ∈ ℝ, λ ⩾ 0\}⋌ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −(1+λ^{2})e_{1}+2λu_{1}+(1+λ^{2})u_{2} & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{1} & (1+λ^{2})e_{1}−2λu_{1}−(1+λ^{2})u_{2} & 0 & 0 & 0 \\ u_{4} & −u_{2} & 0 & −u_{2} & 0 & 0 \end{matrix}

λ ⩾ 0

5.

𝔤̄ = \{( \begin{matrix}λx+(1+μ)y & y & \frac{1+λ^{2}}{μ−1} x+λy \\ −λx−μy & 0 & −\frac{μ+λ^{2}}{μ−1} x−λy \\ x & x & y \end{matrix} )| x, y ∈ ℝ, λ ⩾ 0, μ ≠ 1\} ⋌

ℝ^{3},

𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −\frac{λ^{2}+μ}{μ−1}e_{1}+ \frac{1+λ^{2}}{μ−1} u_{2} & −λe_{1}+u_{1}+λu_{2} \\ u_{2} & 0 & 0 & 0 & −λe_{1}+u_{1}+λu_{2} & −μe_{1}+(μ+1)u_{2} \\ u_{3} & −u_{1} & \frac{λ^{2}+μ}{μ−1}e_{1}− \frac{1+λ^{2}}{μ−1} u_{2} & λe_{1}−u_{1}−λu_{2} & 0 & 0 \\ u_{4} & −u_{2} & λe_{1}−u_{1}−λu_{2} & μe_{1}−(μ+1)u_{2} & 0 & 0 \end{matrix}

λ ⩾ 0, μ ≠ 1

6.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & x \\ −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}⋌[\{( \begin{matrix}0 & y & y \\ −y & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )| y ∈ ℝ\} ⋌ ℝ^{3}],

𝔤 = ⟨( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{2} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{3} & −u_{1} & u_{2} & −u_{1} & 0 & e_{1} \\ u_{4} & −u_{2} & −u_{1} & −u_{2} & −e_{1} & 0 \end{matrix}

7.

𝔤̄ = \{( \begin{matrix}\frac{x+(1+2λ)y}{1+λ} & y & \frac{y−x}{1+λ} \\ −\frac{x+λy}{1+λ} & 0 & \frac{x−y}{1+λ} \\ \frac{x+λy}{1+λ} & x & \frac{y+λx}{1+λ} \end{matrix} )| x, y ∈ ℝ, λ ≠ −1\}⋌ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & x & y \\ u_{2} & 0 & 0 & 0 & y & z \\ u_{3} & −u_{1} & −x & −y & 0 & 0 \\ u_{4} & −u_{2} & −y & −z & 0 & 0 \end{matrix} ,

where

x = \frac{1}{1+λ}e_{1}+ \frac{λ}{1+λ}u_{1}− \frac{1}{1+λ}u_{2},

y = −\frac{1}{1+λ}e_{1}+ \frac{1}{1+λ}u_{1}+ \frac{1}{1+λ}u_{2},

z = −\frac{λ}{1+λ}e_{1}+ \frac{λ}{1+λ}u_{1}+ \frac{1+2λ}{1+λ}u_{2},

λ ≠ −1 8.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & x & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{3} & −u_{1} & 0 & −u_{1} & 0 & −u_{3} \\ u_{4} & −u_{2} & 0 & −u_{2} & u_{3} & 0 \end{matrix}

9.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −λx & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ, λ ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & λu_{1} & −λe_{1}+(λ+1)u_{2} \\ u_{3} & −u_{1} & 0 & −λu_{1} & 0 & −λu_{3} \\ u_{4} & −u_{2} & 0 & λe_{1}−(λ+1)u_{2} & λu_{3} & 0 \end{matrix}

10.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{1} & 0 & 0 & 0 & e_{1} \\ u_{4} & −u_{2} & 0 & −u_{2} & −e_{1} & 0 \end{matrix}

11.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & −x & x & 0 \\ 0 & 0 & x & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & −u_{1} & e_{1} \\ u_{3} & −u_{1} & 0 & u_{1} & 0 & e_{1}+u_{3} \\ u_{4} & −u_{2} & 0 & −e_{1} & −e_{1}−u_{3} & 0 \end{matrix}

12.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & μx & 0 & 0 \\ 0 & 0 & (1 − μ)x & 0 \\ 0 & x & 0 & λx \end{matrix} )| x ∈ ℝ, λ, μ ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 =

⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & μu_{1} & −λμe_{1}+(λ+μ)u_{2} \\ u_{3} & −u_{1} & 0 & −μu_{1} & 0 & (1−μ)u_{3} \\ u_{4} & −u_{2} & −u_{1} & λμe_{1}−(λ+μ)u_{2} & (μ−1)u_{3} & 0 \end{matrix}

EINSTEIN–MAXWELL EQUATION 57

13.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & \frac{x}{2} & x & 0 \\ 0 & 0 & \frac{x}{2} & 0 \\ 0 & x & 0 & λx \end{matrix} )| x ∈ ℝ, λ ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & \frac{1}{2}u_{1} & −\frac{λ}{2}e_{1}+(λ+ \frac{1}{2})u_{2} \\ u_{3} & −u_{1} & 0 & −\frac{1}{2}u_{1} & 0 & e_{1}+ \frac{1}{2}u_{3} \\ u_{4} & −u_{2} & −u_{1} & \frac{λ}{2}e_{1}−(λ+ \frac{1}{2})u_{2} & −e_{1}− \frac{1}{2}u_{3} & 0 \end{matrix}

14.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & (1 − λ)x & x & 0 \\ 0 & 0 & λx & 0 \\ 0 & x & 0 & λx \end{matrix} )| x ∈ ℝ, λ ≠ \frac{1}{2} \}⋌(𝔫_{3} ×ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & (1−λ)u_{1} & λ(λ−1)e_{1}+u_{2} \\ u_{3} & −u_{1} & 0 & (λ−1)u_{1} & 0 & e_{1}+λu_{3} \\ u_{4} & −u_{2} & −u_{1} & λ(1−λ)e_{1}−u_{2} & −e_{1}−λu_{3} & 0 \end{matrix}

λ ≠ \frac{1}{2}

15.

𝔤̄ = \{( \begin{matrix}x & y & y \\ −y & 0 & −x \\ y & x & 2x \end{matrix} )| x, y ∈ ℝ\} ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −e_{1}+2u_{1} & u_{2} \\ u_{2} & 0 & 0 & 0 & u_{2} & −e_{1}+u_{1} \\ u_{3} & −u_{1} & e_{1}−2u_{1} & −u_{2} & 0 & 0 \\ u_{4} & −u_{2} & −u_{2} & e_{1}−u_{1} & 0 & 0 \end{matrix}

16.

𝔤̄ = \{( \begin{matrix}x & y & y \\ y & 0 & −x \\ −y & x & 2x \end{matrix} )| x, y ∈ ℝ\} ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −e_{1}+2u_{1} & u_{2} \\ u_{2} & 0 & 0 & 0 & u_{2} & e_{1}−u_{1} \\ u_{3} & −u_{1} & e_{1}−2u_{1} & −u_{2} & 0 & 0 \\ u_{4} & −u_{2} & −u_{2} & −e_{1}+u_{1} & 0 & 0 \end{matrix}

17.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & x \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{1} \\ u_{3} & −u_{1} & 0 & 0 & 0 & e_{1} \\ u_{4} & −u_{2} & 0 & −u_{1} & −e_{1} & 0 \end{matrix}

18.

𝔤̄ = \{( \begin{matrix}0 & y & x \\ 0 & 0 & y \\ 0 & 0 & 0 \end{matrix} )| x, y ∈ ℝ\} ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & u_{1} \\ u_{3} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{2} & 0 & −u_{1} & 0 & 0 \end{matrix}

19.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & x \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & u_{1} & −e_{1}+u_{1}+2u_{2} \\ u_{3} & −u_{1} & 0 & −u_{1} & 0 & 0 \\ u_{4} & −u_{2} & −u_{1} & e_{1}−u_{1}−2u_{2} & 0 & 0 \end{matrix}

20.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & x & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & x & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & u_{1} & u_{2}−u_{1} \\ u_{3} & −u_{1} & 0 & −u_{1} & 0 & −u_{3} \\ u_{4} & −u_{2} & 0 & u_{1}−u_{2} & u_{3} & 0 \end{matrix}

60 B. KOMRAKOV, JNR.

21.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & (1 − λ)x \\ 0 & λx & 0 & 0 \\ 0 & 0 & (1 − λ)x & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ, λ ≠ 1\} ⋌ (𝔫_{3} × ℝ),

𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & λu_{1} & −λe_{1}+(1−λ)u_{1}+(1+λ)u_{2} \\ u_{3} & −u_{1} & 0 & −λu_{1} & 0 & (1 − λ)u_{3} \\ u_{4} & −u_{2} & −u_{1} & λe_{1}+(λ−1)u_{1}−(1+λ)u_{2} & (λ−1)u_{3} & 0 \end{matrix}

λ ≠ 1

22.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & \frac{x}{2} \\ 0 & \frac{x}{2} & x & 0 \\ 0 & 0 & \frac{x}{2} & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & \frac{1}{2}u_{1} & −\frac{1}{2}e_{1}+ \frac{1}{2}u_{1}+ \frac{3}{2}u_{2} \\ u_{3} & −u_{1} & 0 & −\frac{1}{2}u_{1} & 0 & e_{1} + \frac{1}{2}u_{3} \\ u_{4} & −u_{2} & −u_{1} & \frac{1}{2}e_{1}− \frac{1}{2}u_{1}− \frac{3}{2}u_{2} & −e_{1}− \frac{1}{2}u_{3} & 0 \end{matrix}

23.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & x \\ 0 & 0 & x & 0 \\ 0 & 0 & x & 0 \\ 0 & x & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & 0 & 0 & 0 & u_{1} + u_{2} \\ u_{3} & −u_{1} & 0 & 0 & 0 & e_{1} + u_{3} \\ u_{4} & −u_{2} & −u_{1} & −u_{1}−u_{2} & −e_{1}−u_{3} & 0 \end{matrix}

24.

𝔤̄ = \{( \begin{matrix}λx & y & (2λ − 1)y \\ \frac{(2λ−1)y}{2(λ−1)} & 0 & (1 − 2λ)x \\ −\frac{y}{2(λ−1)} & x & 2λx \end{matrix} )| x, y ∈ ℝ, λ ≠ 1\}⋌ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & (1−2λ)e_{1}+2λu_{1} & (2λ−1)u_{2} \\ u_{2} & 0 & 0 & 0 & λu_{2} & \frac{2λ−1}{2(λ−1)}e_{1}− \frac{1}{2(λ−1)}u_{1} \\ u_{3} & −u_{1} & (2λ−1)e_{1}−2λu_{1} & −λu_{2} & 0 & (λ − 1)u_{4} \\ u_{4} & −u_{2} & (1−2λ)u_{2} & \frac{1−2λ}{2(λ−1)}e_{1}+ \frac{1}{2(λ−1)}u_{1} & (1−λ)u_{4} & 0 \end{matrix}

λ ≠ 1

25.

𝔤̄ = \{( \begin{matrix}λx & y & (2λ − 1)y \\ \frac{(1−2λ)y}{2(λ−1)} & 0 & (1 − 2λ)x \\ \frac{y}{2(λ−1)} & x & 2λx \end{matrix} )| x, y ∈ ℝ, λ ≠ 1\}⋌ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & (1−2λ)e_{1}+2λu_{1} & (2λ−1)u_{2} \\ u_{2} & 0 & 0 & 0 & λu_{2} & \frac{1−2λ}{2(λ−1)}e_{1}+ \frac{1}{2(λ−1)}u_{1} \\ u_{3} & −u_{1} & (2λ−1)e_{1}−2λu_{1} & −λu_{2} & 0 & (λ − 1)u_{4} \\ u_{4} & −u_{2} & (1−2λ)u_{2} & −\frac{1−2λ}{2(λ−1)}e_{1}− \frac{1}{2(λ−1)}u_{1} & (1−λ)u_{4} & 0 \end{matrix}

λ ≠ 1

62 B. KOMRAKOV, JNR.

26.

𝔤̄ = \{( \begin{matrix}\frac{x}{3} & 0 & 0 & 0 \\ x & 0 & −\frac{x}{3} & 0 \\ 0 & x & \frac{4}{3}x & 0 \\ 0 & 0 & 0 & \frac{2}{3}x \end{matrix} )| x ∈ ℝ\}⋌[\{( \begin{matrix}0 & 0 & −\frac{y}{2} \\ 0 & 0 & \frac{3}{2}y \\ y & \frac{y}{3} & 0 \end{matrix} )| y ∈ ℝ\} ⋌ ℝ^{3}],

𝔤 = ⟨( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −\frac{1}{3}e_{1}+ \frac{4}{3}u_{1} & \frac{1}{3}u_{2} \\ u_{2} & 0 & 0 & 0 & \frac{2}{3}u_{2} & −\frac{1}{2}e_{1}+ \frac{3}{2}u_{1} \\ u_{3} & −u_{1} & \frac{1}{3}e_{1}− \frac{4}{3}u_{1} & −\frac{2}{3}u_{2} & 0 & e_{1}− \frac{1}{3}u_{4} \\ u_{4} & −u_{2} & −\frac{1}{3}u_{2} & \frac{1}{2}e_{1}− \frac{3}{2}u_{1} & \frac{1}{3}u_{4}−e_{1} & 0 \end{matrix}

27.

𝔤̄ = \{( \begin{matrix}\frac{x}{3} & 0 & 0 & 0 \\ x & 0 & −\frac{x}{3} & 0 \\ 0 & x & \frac{4}{3}x & 0 \\ 0 & 0 & 0 & \frac{2}{3}x \end{matrix} )| x ∈ ℝ\}⋌[\{( \begin{matrix}0 & 0 & \frac{y}{2} \\ 0 & 0 & −\frac{3}{2}y \\ y & \frac{y}{3} & 0 \end{matrix} )| y ∈ ℝ\} ⋌ ℝ^{3}],

𝔤 = ⟨( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & −\frac{1}{3}e_{1}+ \frac{4}{3}u_{1} & \frac{1}{3}u_{2} \\ u_{2} & 0 & 0 & 0 & \frac{2}{3}u_{2} & \frac{1}{2}e_{1}− \frac{3}{2}u_{1} \\ u_{3} & −u_{1} & \frac{1}{3}e_{1}− \frac{4}{3}u_{1} & −\frac{2}{3}u_{2} & 0 & e_{1}− \frac{1}{3}u_{4} \\ u_{4} & −u_{2} & −\frac{1}{3}u_{2} & −\frac{1}{2}e_{1}+ \frac{3}{2}u_{1} & \frac{1}{3}u_{4}−e_{1} & 0 \end{matrix}

28.

𝔤̄ = \{( \begin{matrix}0 & 0 & y \\ x & 2x & −\frac{y}{2} \\ y & 2y & x \end{matrix} )| x, y ∈ ℝ\} ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 2u_{1} & 2u_{2} \\ u_{2} & 0 & 0 & 0 & u_{2} & e_{1}− \frac{1}{2}u_{1} \\ u_{3} & −u_{1} & −2u_{1} & −u_{2} & 0 & u_{4} \\ u_{4} & −u_{2} & −2u_{2} & \frac{1}{2}u_{1}−e_{1} & −u_{4} & 0 \end{matrix}

29.

𝔤̄ = \{( \begin{matrix}0 & 0 & −y \\ x & 2x & \frac{y}{2} \\ y & 2y & x \end{matrix} )| x, y ∈ ℝ\} ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 2u_{1} & 2u_{2} \\ u_{2} & 0 & 0 & 0 & u_{2} & −e_{1}+ \frac{1}{2}u_{1} \\ u_{3} & −u_{1} & −2u_{1} & −u_{2} & 0 & u_{4} \\ u_{4} & −u_{2} & −2u_{2} & e_{1}− \frac{1}{2}u_{1} & −u_{4} & 0 \end{matrix}

30.

𝔤̄ = \{( \begin{matrix}b_{3}x + c_{3}y & y & a_{3}x + b_{3}y \\ b_{1}x + c_{1}y & 0 & a_{1}x + b_{1}y \\ b_{2}x + c_{2}y & x & a_{2}x + b_{2}y \end{matrix} )| x, y ∈ ℝ\}⋌ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} )⟩ ,

where

a_{1} = \frac{λμ(λ − 1)}{λ + μ − λμ},  a_{2} = \frac{λ^{2} + μ − λ^{2}μ}{λ + μ − λμ},  a_{3} = \frac{λ(1 − λ)}{λ + μ − λμ},
b_{1} = −\frac{λμ}{λ + μ − λμ},  b_{2} = \frac{μ}{λ + μ − λμ},  b_{3} = \frac{λ}{λ + μ − λμ},
c_{1} = \frac{λμ(μ − 1)}{λ + μ − λμ},  c_{2} = \frac{μ(1 − μ)}{λ + μ − λμ},  c_{3} = \frac{λ + μ^{2} − μ^{2}λ}{λ + μ − λμ},

λ + μ − λμ ≠ 0, −1 ⩽ μ ⩽ λ, λμ > 0

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & x & y \\ u_{2} & 0 & 0 & 0 & y & z \\ u_{3} & −u_{1} & −x & −y & 0 & 0 \\ u_{4} & −u_{2} & −y & −z & 0 & 0 \end{matrix} ,

where

x = \frac{λμ(λ − 1)}{λ + μ − λμ}e_{1} + \frac{λ^{2} + μ − λ^{2}μ}{λ + μ − λμ}u_{1} + \frac{λ(1 − λ)}{λ + μ − λμ}u_{2},

y = −\frac{λμ}{λ + μ − λμ}e_{1} + \frac{μ}{λ + μ − λμ}u_{1} + \frac{λ}{λ + μ − λμ}u_{2},

z = \frac{λμ(μ − 1)}{λ + μ − λμ}e_{1} + \frac{μ(1 − μ)}{λ + μ − λμ}u_{1} + \frac{λ + μ^{2} − μ^{2}λ}{λ + μ − λμ}u_{2},

λ + μ − λμ ≠ 0, −1 ⩽ μ ⩽ λ, λμ > 0

31. 𝔤̄ = 𝔫̃_{5}, 𝔤 = ⟨h⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & −u_{1} & 0 & 0 & 0 & e_{1} \\ u_{4} & −u_{2} & 0 & 0 & −e_{1} & 0 \end{matrix}

32. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 1.3^{1}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{2} & 0 & 0 & 0 & 0 \end{matrix}

1.4^{1}

𝔤 = \{( \begin{matrix}0 & x & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}

1.

𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔯_{2}, 𝔤 = ⟨( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) + p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & e_{1} \\ u_{1} & 0 & 0 & u_{1} & u_{2} & u_{1} \\ u_{2} & −u_{1} & −u_{1} & 0 & u_{3} & 0 \\ u_{3} & −u_{2} & −u_{2} & −u_{3} & 0 & −u_{3} \\ u_{4} & −e_{1} & −u_{1} & 0 & u_{3} & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & px & 0 & 0 \\ 0 & 0 & (p − 1)x & 0 \\ 0 & 0 & 0 & (p − 2)x \end{matrix} )| x ∈ ℝ, p ∈ ℝ\} ⋌ 𝔫_{4}, 𝔤 = ⟨h⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & e_{1} \\ u_{1} & 0 & 0 & 0 & 0 & pu_{1} \\ u_{2} & −u_{1} & 0 & 0 & 0 & (p−1)u_{2} \\ u_{3} & −u_{2} & 0 & 0 & 0 & (p−2)u_{3} \\ u_{4} & −e_{1} & −pu_{1} & (1−p)u_{2} & (2−p)u_{3} & 0 \end{matrix}

3.

𝔤̄ = \{( \begin{matrix}2y & 0 & 0 \\ 0 & y & x \\ 0 & x & y \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{3}, 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & e_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 2u_{1} \\ u_{2} & −u_{1} & 0 & 0 & e_{1} & u_{2} \\ u_{3} & −u_{2} & 0 & −e_{1} & 0 & 0 \\ u_{4} & −e_{1} & −2u_{1} & −u_{2} & 0 & 0 \end{matrix}

4.

𝔤̄ = \{( \begin{matrix}2y & 0 & 0 \\ 0 & y & −x \\ 0 & x & y \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{3}, 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & e_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 2u_{1} \\ u_{2} & −u_{1} & 0 & 0 & −e_{1} & u_{2} \\ u_{3} & −u_{2} & 0 & e_{1} & 0 & 0 \\ u_{4} & −e_{1} & −2u_{1} & −u_{2} & 0 & 0 \end{matrix}

5.

𝔤̄ = 𝔰𝔩(2, ℝ) × ℝ × ℝ, 𝔤 = \{[( \begin{matrix}0 & 0 \\ z & 0 \end{matrix} ) , z, 0]| z ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{2} & −u_{1} & −u_{1} & 0 & u_{3} & 0 \\ u_{3} & −u_{2} & −u_{2} & −u_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

6.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & x & 0 & x \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{4}, 𝔤 = ⟨h⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & −u_{1} & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{2} & 0 & 0 & 0 & u_{1}+u_{3} \\ u_{4} & 0 & −u_{1} & −u_{2} & −u_{1}−u_{3} & 0 \end{matrix}

7.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & x & 0 & −x \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{4}, 𝔤 = ⟨h⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & −u_{1} & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{2} & 0 & 0 & 0 & −u_{1}+u_{3} \\ u_{4} & 0 & −u_{1} & −u_{2} & u_{1}−u_{3} & 0 \end{matrix}

8.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & 0 & 0 & x \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{4}, 𝔤 = ⟨h⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & −u_{1} & 0 & 0 & 0 & u_{2} \\ u_{3} & −u_{2} & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & −u_{1} & −u_{2} & −u_{3} & 0 \end{matrix}

9.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & rx & 0 \\ 0 & x & x & 0 \\ 0 & 0 & x & −px \end{matrix} )| x ∈ ℝ, p, r ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & re_{1}+u_{2}+u_{4} & 0 \\ u_{3} & −u_{2} & −u_{1} & −re_{1}−u_{2}−u_{4} & 0 & pu_{4} \\ u_{4} & 0 & 0 & 0 & −pu_{4} & 0 \end{matrix}

10.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & rx & 0 \\ 0 & x & x & 0 \\ 0 & 0 & 0 & −px \end{matrix} )| x ∈ ℝ, p, r ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & re_{1}+u_{2} & 0 \\ u_{3} & −u_{2} & −u_{1} & −re_{1}−u_{2} & 0 & pu_{4} \\ u_{4} & 0 & 0 & 0 & −pu_{4} & 0 \end{matrix}

11.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & −x \\ 0 & 0 & rx & 0 \\ 0 & x & x & 0 \\ 0 & 0 & x & x \end{matrix} )| x ∈ ℝ, r ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & re_{1}+u_{2}+u_{4} & 0 \\ u_{3} & −u_{2} & −u_{1} & −re_{1}−u_{2}−u_{4} & 0 & u_{1}−u_{4} \\ u_{4} & 0 & 0 & 0 & u_{4}−u_{1} & 0 \end{matrix}

12.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & −x \\ 0 & 0 & rx & 0 \\ 0 & x & x & 0 \\ 0 & 0 & 0 & x \end{matrix} )| x ∈ ℝ, r ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & re_{1}+u_{2} & 0 \\ u_{3} & −u_{2} & −u_{1} & −re_{1}−u_{2} & 0 & u_{1}−u_{4} \\ u_{4} & 0 & 0 & 0 & u_{4}−u_{1} & 0 \end{matrix}

13.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & rx & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & −x \end{matrix} )| x ∈ ℝ, r ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & re_{1}+u_{4} & 0 \\ u_{3} & −u_{2} & 0 & −re_{1}−u_{4} & 0 & u_{4} \\ u_{4} & 0 & 0 & 0 & −u_{4} & 0 \end{matrix}

14.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & rx & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & −x \end{matrix} )| x ∈ ℝ, r ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & re_{1} & 0 \\ u_{3} & −u_{2} & 0 & −re_{1} & 0 & u_{4} \\ u_{4} & 0 & 0 & 0 & −u_{4} & 0 \end{matrix}

15.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & 0 & x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & e_{1}+u_{4} & 0 \\ u_{3} & −u_{2} & 0 & −e_{1}−u_{4} & 0 & u_{1} \\ u_{4} & 0 & 0 & 0 & −u_{1} & 0 \end{matrix}

16.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & 0 & −x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & −e_{1}+u_{4} & 0 \\ u_{3} & −u_{2} & 0 & e_{1}−u_{4} & 0 & u_{1} \\ u_{4} & 0 & 0 & 0 & −u_{1} & 0 \end{matrix}

17.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & u_{4} & 0 \\ u_{3} & −u_{2} & 0 & −u_{4} & 0 & u_{1} \\ u_{4} & 0 & 0 & 0 & −u_{1} & 0 \end{matrix}

18.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & e_{1}+u_{4} & 0 \\ u_{3} & −u_{2} & 0 & −e_{1}−u_{4} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

19.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & −e_{1}+u_{4} & 0 \\ u_{3} & −u_{2} & 0 & e_{1}−u_{4} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

20.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & u_{4} & 0 \\ u_{3} & −u_{2} & 0 & −u_{4} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

21.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & 0 & x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & e_{1} & 0 \\ u_{3} & −u_{2} & 0 & −e_{1} & 0 & u_{1} \\ u_{4} & 0 & 0 & 0 & −u_{1} & 0 \end{matrix}

22.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & 0 & −x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & −e_{1} & 0 \\ u_{3} & −u_{2} & 0 & e_{1} & 0 & u_{1} \\ u_{4} & 0 & 0 & 0 & −u_{1} & 0 \end{matrix}

23.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & −x \\ 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & −u_{2} & 0 & 0 & 0 & u_{1} \\ u_{4} & 0 & 0 & 0 & −u_{1} & 0 \end{matrix}

24.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & e_{1} & 0 \\ u_{3} & −u_{2} & 0 & −e_{1} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

25.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ (𝔫_{3} × ℝ), 𝔤 = ⟨p⟩

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & −e_{1} & 0 \\ u_{3} & −u_{2} & 0 & e_{1} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & −u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

2.1^{1}

𝔤 = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & y & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & −y \end{matrix} )| x, y ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔰𝔩(2, ℝ), 𝔤 = 𝔰𝔬(1, 1) × 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{2} & 0 & −u_{4} \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{2} & 0 & 0 & 0 & e_{2} \\ u_{3} & u_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{4} & 0 & −e_{2} & 0 & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔩(2, ℝ) × (𝔰𝔬(1, 1) ⋌ ℝ^{2}), 𝔤 = 𝔰𝔬(1, 1) × 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{2} & 0 & −u_{4} \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{2} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{4} & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{2} & 0 & −u_{4} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{2} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{4} & 0 & 0 & 0 & 0 \end{matrix}

2.1^{2}

𝔤 = \{( \begin{matrix}x & 0 & 0 & 0 \\ 0 & 0 & 0 & −y \\ 0 & 0 & −x & 0 \\ 0 & y & 0 & 0 \end{matrix} )| x, y ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔰𝔲(2), 𝔤 = 𝔰𝔬(1, 1) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & e_{2} \\ u_{3} & u_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & −e_{2} & 0 & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔰𝔩(2, ℝ), 𝔤 = 𝔰𝔬(1, 1) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & −e_{2} \\ u_{3} & u_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & e_{2} & 0 & 0 \end{matrix}

3. 𝔤̄ = 𝔰𝔩(2, ℝ) × (𝔰𝔬(2) ⋌ ℝ^{2}), 𝔤 = 𝔰𝔬(1, 1) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \end{matrix}

4. 𝔤̄ = 𝔰𝔲(2) × (𝔰𝔬(1, 1) ⋌ ℝ^{2}), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & e_{2} \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & −e_{2} & 0 & 0 \end{matrix}

5. 𝔤̄ = 𝔰𝔩(2, ℝ) × (𝔰𝔬(1, 1) ⋌ ℝ^{2}), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(1, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & −e_{2} \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & e_{2} & 0 & 0 \end{matrix}

6. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.1^{2}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \end{matrix}

2.1^{3}

𝔤 = \{( \begin{matrix}0 & 0 & −x & 0 \\ 0 & 0 & 0 & −y \\ x & 0 & 0 & 0 \\ 0 & y & 0 & 0 \end{matrix} )| x, y ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔲(2) × 𝔰𝔲(2), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{3} & 0 & −u_{1} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & e_{2} \\ u_{3} & u_{1} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & −e_{2} & 0 & 0 \end{matrix}

2.

𝔤̄ = 𝔰𝔲(2) × 𝔰𝔩(2, ℝ), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{3} & 0 & −u_{1} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & −e_{2} \\ u_{3} & u_{1} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & e_{2} & 0 & 0 \end{matrix}

3.

𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔰𝔩(2, ℝ), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{3} & 0 & −u_{1} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & −e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & −e_{2} \\ u_{3} & u_{1} & 0 & e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & e_{2} & 0 & 0 \end{matrix}

4.

𝔤̄ = 𝔰𝔲(2) × (𝔰𝔬(2) ⋌ ℝ^{2}), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{3} & 0 & −u_{1} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & 0 & −e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \end{matrix}

5. 𝔤̄ = 𝔰𝔩(2, ℝ) × (𝔰𝔬(2) ⋌ ℝ^{2}), 𝔤 = 𝔰𝔬(2) × 𝔰𝔬(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{3} & 0 & −u_{1} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & −e_{1} & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & 0 & e_{1} & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \end{matrix}

6. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.1^{3}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{3} & 0 & −u_{1} & 0 \\ e_{2} & 0 & 0 & 0 & u_{4} & 0 & −u_{2} \\ u_{1} & −u_{3} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{4} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \end{matrix}

2.1^{4}

𝔤 = \{( \begin{matrix}x & −y & 0 & 0 \\ y & x & 0 & 0 \\ 0 & 0 & −x & y \\ 0 & 0 & −y & −x \end{matrix} )| x, y ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔩(2, ℂ)_{ℝ}, 𝔤 = 𝔰𝔬(2, ℂ)_{ℝ}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{2} & 0 & 0 & u_{2} & −u_{1} & −u_{4} & u_{3} \\ u_{1} & −u_{1} & −u_{2} & 0 & 0 & e_{1} & e_{2} \\ u_{2} & −u_{2} & u_{1} & 0 & 0 & e_{2} & −e_{1} \\ u_{3} & u_{3} & u_{4} & −e_{1} & −e_{2} & 0 & 0 \\ u_{4} & u_{4} & −u_{3} & −e_{2} & e_{1} & 0 & 0 \end{matrix}

2.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.1^{4}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{2} & 0 & 0 & u_{2} & −u_{1} & −u_{4} & u_{3} \\ u_{1} & −u_{1} & −u_{2} & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & u_{4} & −u_{3} & 0 & 0 & 0 & 0 \end{matrix}

2.2^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & 0 \\ 0 & λx & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & −λx \end{matrix} )| x, y ∈ ℝ, λ ∈ [−1, 1]\}

λ = 0 1.

𝔤̄ = 𝔤𝔩(2, ℝ) ⋌ ℝ^{2}, 𝔤 = \{( \begin{matrix}x & y \\ 0 & 0 \end{matrix} )| x, y ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & u_{1} & −u_{4} & −2e_{2} \\ u_{1} & −u_{1} & 0 & 0 & 0 & u_{2} & −u_{1} \\ u_{2} & 0 & −u_{1} & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{3} & u_{4} & −u_{2} & 0 & 0 & 2u_{3} \\ u_{4} & 0 & 2e_{2} & u_{1} & −u_{2} & −2u_{3} & 0 \end{matrix}

EINSTEIN–MAXWELL EQUATION 81

2.

𝔤̄ = \{( \begin{matrix}x − y & 0 & 0 & −y \\ 0 & py & 0 & 0 \\ 0 & 0 & (p − 1)y − x & 0 \\ 0 & 0 & 0 & x + y \end{matrix} )| x, y ∈ ℝ, p ∈ ℝ\}⋌(ℝ×

𝔫_{3}),

𝔤 = ⟨( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , q⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & e_{2} & u_{4} & 0 \\ u_{2} & 0 & −u_{1} & −e_{2} & 0 & (p−1)u_{3} & pu_{4} \\ u_{3} & u_{3} & u_{4} & −u_{4} & (1−p)u_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & −pu_{4} & 0 & 0 \end{matrix}

3.

𝔤̄ = 𝔤 ⋌ (ℝ × (ℝ id_{ℝ^{2}} ⋌ℝ^{2})), 𝔤 = \{( \begin{matrix}x & y & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & 0 \end{matrix} )| x, y ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{1} & 0 & 0 & u_{3} & u_{4} \\ u_{3} & u_{3} & u_{4} & 0 & −u_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & −u_{4} & 0 & 0 \end{matrix}

λ = 1

4.

𝔤̄ = 𝔰𝔬(2, 1) ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{2} & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & e_{1} & e_{2} \\ u_{3} & u_{3} & u_{4} & −e_{2} & −e_{1} & 0 & 0 \\ u_{4} & u_{4} & 0 & 0 & −e_{2} & 0 & 0 \end{matrix}

5.

𝔤̄ = \{( \begin{matrix}−x & 0 & y & 0 \\ 0 & x & 0 & 0 \\ 0 & 0 & 0 & −y \\ 0 & 0 & 0 & x \end{matrix} )| x, y ∈ ℝ\}⋌(ℝ×𝔫_{3}), 𝔤 = ⟨( \begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , p⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & e_{2} & 0 \\ u_{3} & u_{3} & u_{4} & 0 & −e_{2} & 0 & 0 \\ u_{4} & u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

λ = −\frac{1}{2} 6.

𝔤̄ = 𝔤 ⋌ (𝔫_{3} × ℝ), 𝔤 = \{( \begin{matrix}\frac{x}{2} & 0 & 0 & −y \\ 0 & x & y & 0 \\ 0 & 0 & −\frac{x}{2} & 0 \\ 0 & 0 & 0 & −x \end{matrix} )| x, y ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & \frac{3}{2}e_{2} & u_{1} & −\frac{1}{2}u_{2} & −u_{3} & \frac{1}{2}u_{4} \\ e_{2} & −\frac{3}{2}e_{2} & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & u_{4} & 0 & 0 \\ u_{2} & \frac{1}{2}u_{2} & −u_{1} & −u_{4} & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & −\frac{1}{2}u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

λ ∈ [ −1, 1]

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & (1−λ)e_{2} & u_{1} & λu_{2} & −u_{3} & −λu_{4} \\ e_{2} & (λ−1)e_{2} & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −λu_{2} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & λu_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

2.2^{2}

𝔤 = \{( \begin{matrix}0 & −x & y & 0 \\ x & 0 & 0 & y \\ 0 & 0 & 0 & −x \\ 0 & 0 & x & 0 \end{matrix} )| x, y ∈ ℝ\}

1.

𝔤̄ = 𝔰𝔬(2, 1) ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & 1 & 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{2} & −u_{1} & u_{4} & −u_{3} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & −u_{2} & 0 & 0 & 0 & e_{2} & 0 \\ u_{2} & u_{1} & 0 & 0 & 0 & 0 & e_{2} \\ u_{3} & −u_{4} & −u_{1} & −e_{2} & 0 & 0 & −e_{1} \\ u_{4} & u_{3} & −u_{2} & 0 & −e_{2} & e_{1} & 0 \end{matrix}

2.

𝔤̄ = 𝔰𝔬(3) ⋌ ℝ^{3}, 𝔤 = ⟨( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & 1 & 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} )⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{2} & −u_{1} & u_{4} & −u_{3} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & −u_{2} & 0 & 0 & 0 & −e_{2} & 0 \\ u_{2} & u_{1} & 0 & 0 & 0 & 0 & −e_{2} \\ u_{3} & −u_{4} & −u_{1} & e_{2} & 0 & 0 & e_{1} \\ u_{4} & u_{3} & −u_{2} & 0 & e_{2} & −e_{1} & 0 \end{matrix}

3.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −x \\ 0 & 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\}⋌𝔫̃_{5}, 𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 1 & 0 \end{matrix} ) , h⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{2} & −u_{1} & u_{4} & −u_{3} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & −u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & −u_{4} & −u_{1} & 0 & 0 & 0 & e_{2} \\ u_{4} & u_{3} & −u_{2} & 0 & 0 & −e_{2} & 0 \end{matrix}

4.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.2^{2}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & u_{2} & −u_{1} & u_{4} & −u_{3} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & −u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & −u_{4} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & u_{3} & −u_{2} & 0 & 0 & 0 & 0 \end{matrix}

2.2^{3}

𝔤 = \{( \begin{matrix}−x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} & y & 0 \\ −x cos \frac{ϕ}{2} & −x sin \frac{ϕ}{2} & 0 & y \\ 0 & 0 & x sin \frac{ϕ}{2} & x cos \frac{ϕ}{2} \\ 0 & 0 & −x cos \frac{ϕ}{2} & x sin \frac{ϕ}{2} \end{matrix} )| x, y ∈ ℝ, ϕ ∈]0, π[\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.2^{3}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −2 sin \frac{ϕ}{2}e_{2} & A & B & C & D \\ e_{2} & 2 sin \frac{ϕ}{2}e_{2} & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & −A & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −B & 0 & 0 & 0 & 0 & 0 \\ u_{3} & −C & −u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & −D & −u_{2} & 0 & 0 & 0 & 0 \end{matrix} ,

where A = − sin \frac{ϕ}{2}u_{1} − cos \frac{ϕ}{2}u_{2}, B = cos \frac{ϕ}{2}u_{1} − sin \frac{ϕ}{2}u_{2}, C = sin \frac{ϕ}{2}u_{3} − cos \frac{ϕ}{2}u_{4}, D = cos \frac{ϕ}{2}u_{3} + sin \frac{ϕ}{2}u_{4}

2.3^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & x \\ 0 & −x & −x & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & x \end{matrix} )| x, y ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.3^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & u_{1} & −u_{2} & −u_{2}−u_{3} & u_{1}+u_{4} \\ e_{2} & −2e_{2} & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & u_{2}+u_{3} & u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{1}−u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

𝔤 = \{( \begin{matrix}x & y & 0 & 0 \\ 0 & 0 & y & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x, y ∈ ℝ\}

1.

𝔤̄ = 𝔰𝔩(2, ℝ) × 𝔯_{2} × ℝ, 𝔤 = ⟨( \begin{matrix}\frac{1}{2} & 0 \\ 0 & −\frac{1}{2} \end{matrix} ) − q, ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) + p⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & −u_{1} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{2} & 0 & −u_{1} & −u_{1} & 0 & u_{3} & 0 \\ u_{3} & u_{3} & −u_{2} & −u_{2} & −u_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

2.

𝔤̄ = (𝔤 ⊕ ℝ id_{ℝ^{3}}) ⋌ ℝ^{3}, 𝔤 = \{( \begin{matrix}x & y & 0 \\ 0 & 0 & y \\ 0 & 0 & −x \end{matrix} )| x, y ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & −u_{1} & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{3} & −u_{2} & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & 0 & −u_{1} & −u_{2} & −u_{3} & 0 \end{matrix}

3.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.4^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & u_{1} & u_{2} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & −u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

2.5^{1}

𝔤 = \{( \begin{matrix}0 & x & 0 & y \\ 0 & 0 & −y & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 \end{matrix} )| x, y ∈ ℝ\}

1.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 \\ 0 & x & y \\ 0 & z & −x \end{matrix} )| x, y, z ∈ ℝ\}⋌𝔫_{3}, 𝔤 = ⟨( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) + h, p⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & −2e_{1} \\ e_{2} & 0 & 0 & 0 & −2e_{2} & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 2e_{2}−u_{1} & u_{2}+u_{4} & 2e_{1}−u_{1} \\ u_{2} & −u_{1} & 2e_{2} & u_{1}−2e_{2} & 0 & −2u_{3} & u_{2}−u_{4} \\ u_{3} & u_{4} & u_{2} & −u_{2}−u_{4} & 2u_{3} & 0 & 2u_{3} \\ u_{4} & 2e_{1} & −u_{1} & u_{1}−2e_{1} & u_{4}−u_{2} & −2u_{3} & 0 \end{matrix}

2.

𝔤̄ = (𝔰𝔩(2, ℝ) ⋌ ℝ^{2}) × ℝ, 𝔤 = \{[( \begin{matrix}0 & y \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}−t \\ 0 \end{matrix} ) , t]| y, t ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & −2e_{2} & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & −u_{1} & u_{4} & 0 \\ u_{2} & −u_{1} & 2e_{2} & u_{1} & 0 & −2u_{3} & −u_{4} \\ u_{3} & u_{4} & u_{2} & −u_{4} & 2u_{3} & 0 & 0 \\ u_{4} & 0 & −u_{1} & 0 & u_{4} & 0 & 0 \end{matrix}

3.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & gx \\ 0 & 0 & −hx & gx & −kx \\ 0 & 0 & −x & x & 0 \\ 0 & −x & 0 & 0 & (1 + h)x \end{matrix} )| \begin{matrix}x ∈ ℝ, g, k ∈ ℝ, \\ h ⩾ 0 (if k ≠ 0), \\ h ∈ ℝ (if k = 0) \end{matrix}\} ⋌ 𝔫_{5},

𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1}+ge_{2}+(1−h)u_{2} & hu_{1} \\ u_{3} & u_{4} & u_{2} & −u_{1} & −e_{1}−ge_{2}+(h−1)u_{2} & 0 & −(g+h)e_{1}+ke_{2}−(1+h)u_{4} \\ u_{4} & 0 & −u_{1} & 0 & −hu_{1} & (g+h)e_{1}−ke_{2}+(1+h)u_{4} & 0 \end{matrix}

h ⩾ 0 (if k ≠ 0), h ∈ ℝ (if k = 0)

4.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & gx \\ 0 & 0 & −hx & gx & 0 \\ 0 & 0 & −x & x & 0 \\ 0 & −x & 0 & 0 & (1+h)x \end{matrix} )| x ∈ ℝ, g ∈ ℝ, h ⩾ 0\} ⋌ 𝔫_{5},

𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & ge_{2}+(1−h)u_{2} & hu_{1} \\ u_{3} & u_{4} & u_{2} & −u_{1} & −ge_{2}+(h−1)u_{2} & 0 & −(g+h)e_{1}−(1+h)u_{4} \\ u_{4} & 0 & −u_{1} & 0 & −hu_{1} & (g+h)e_{1}+(1+h)u_{4} & 0 \end{matrix}

h ⩾ 0

5.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & gx \\ 0 & 0 & −x & gx & −kx \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & x \end{matrix} )| x ∈ ℝ, g, k ∈ ℝ\}⋌𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1}+ge_{2}−u_{2} & u_{1} \\ u_{3} & u_{4} & u_{2} & 0 & −e_{1}−ge_{2}+u_{2} & 0 & −ge_{1}+ke_{2}−u_{4} \\ u_{4} & 0 & −u_{1} & 0 & −u_{1} & ge_{1}−ke_{2}+u_{4} & 0 \end{matrix}

6.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & gx \\ 0 & 0 & −x & gx & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & x \end{matrix} )| x ∈ ℝ, g ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & ge_{2}−u_{2} & u_{1} \\ u_{3} & u_{4} & u_{2} & 0 & −ge_{2}+u_{2} & 0 & −ge_{1}−u_{4} \\ u_{4} & 0 & −u_{1} & 0 & −u_{1} & ge_{1}+u_{4} & 0 \end{matrix}

7.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & x \\ 0 & 0 & 0 & x & −kx \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ, k ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1}+e_{2} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & −e_{1}−e_{2} & 0 & −e_{1}+ke_{2} \\ u_{4} & 0 & −u_{1} & 0 & 0 & e_{1}−ke_{2} & 0 \end{matrix}

8.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & −x \\ 0 & 0 & 0 & −x & −kx \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ, k ∈ ℝ\}⋌𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1}−e_{2} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & −e_{1}+e_{2} & 0 & e_{1}+ke_{2} \\ u_{4} & 0 & −u_{1} & 0 & 0 & −e_{1}−ke_{2} & 0 \end{matrix}

9.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & 0 & 0 & x & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{2} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & −e_{2} & 0 & −e_{1} \\ u_{4} & 0 & −u_{1} & 0 & 0 & e_{1} & 0 \end{matrix}

10.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −x \\ 0 & 0 & 0 & −x & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & −e_{2} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & e_{2} & 0 & e_{1} \\ u_{4} & 0 & −u_{1} & 0 & 0 & −e_{1} & 0 \end{matrix}

11.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & −x \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & −e_{1} & 0 & e_{2} \\ u_{4} & 0 & −u_{1} & 0 & 0 & −e_{2} & 0 \end{matrix}

12.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & −e_{1} & 0 & −e_{2} \\ u_{4} & 0 & −u_{1} & 0 & 0 & e_{2} & 0 \end{matrix}

13.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{1} & 0 \\ u_{3} & u_{4} & u_{2} & 0 & −e_{1} & 0 & 0 \\ u_{4} & 0 & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{4} & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

2.5^{2}

𝔤 = \{( \begin{matrix}0 & x & 0 & −y \\ 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & y & 0 \end{matrix} )| x, y ∈ ℝ\}

1.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 \\ 0 & x & y \\ 0 & z & −x \end{matrix} )| x, y, z ∈ ℝ\}⋌𝔫_{3}, 𝔤 = ⟨( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) − h, p⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & −e_{1} + u_{1} & −u_{2} & e_{2} \\ e_{2} & 0 & 0 & 0 & −e_{2} & u_{4} & −e_{1} − u_{1} \\ u_{1} & 0 & 0 & 0 & e_{1} − u_{1} & u_{2} & −e_{2} \\ u_{2} & e_{1} − u_{1} & e_{2} & −e_{1} + u_{1} & 0 & −2u_{3} & −u_{4} \\ u_{3} & u_{2} & −u_{4} & −u_{2} & 2u_{3} & 0 & 0 \\ u_{4} & −e_{2} & e_{1} + u_{1} & e_{2} & u_{4} & 0 & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 0 & rx & (p+s)x & 0 \\ 0 & 0 & x & 0 & x \\ 0 & −x & 2rx & x & 0 \\ 0 & 2rx & (s−p−4r^{2})x & −rx & 0 \end{matrix} )| \begin{matrix}x ∈ ℝ, p ∈ ℝ, \\ r, s ⩾ 0 \end{matrix}\}⋌𝔫_{5},

𝔤 = ⟨p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & A & 2ru_{1} \\ u_{3} & u_{2} & −u_{4} & −u_{1} & −A & 0 & B \\ u_{4} & 0 & u_{1} & 0 & −2ru_{1} & −B & 0 \end{matrix} ,

where

\{ \begin{matrix}A = (p + s)e_{1} + re_{2} + u_{2} − 2ru_{4}, \\ B = −re_{1} + (p − s)e_{2} − 2ru_{2} − u_{4}, \\ r ⩾ 0, s ⩾ 0 \end{matrix}

3.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −(r+s)x & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & −x & x & 0 & 0 \\ 0 & x & (r−s−1)x & 0 & 0 \end{matrix} )| x ∈ ℝ, r ∈ ℝ, s ⩾ 0\}⋌

𝔫_{5},

𝔤 = ⟨p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & −(r + s)e_{1} − u_{4} & u_{1} \\ u_{3} & u_{2} & −u_{4} & 0 & (r + s)e_{1} + u_{4} & 0 & (s − r)e_{2} − u_{2} \\ u_{4} & 0 & u_{1} & 0 & −u_{1} & (r − s)e_{2} + u_{2} & 0 \end{matrix}

s ⩾ 0

4.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & (1 + s)x & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & −x & 0 & 0 & 0 \\ 0 & 0 & (s − 1)x & 0 & 0 \end{matrix} )| x ∈ ℝ, s ⩾ 0\} ⋌ 𝔫_{5}, 𝔤 =

⟨p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & (1 + s)e_{1} & 0 \\ u_{3} & u_{2} & −u_{4} & 0 & −(1 + s)e_{1} & 0 & (1 − s)e_{2} \\ u_{4} & 0 & u_{1} & 0 & 0 & (s − 1)e_{2} & 0 \end{matrix}

s ⩾ 0

5.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −(1+s)x & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & −x & 0 & 0 & 0 \\ 0 & 0 & (1−s)x & 0 & 0 \end{matrix} )| x ∈ ℝ, s ⩾ 0\} ⋌ 𝔫_{5}, 𝔤 =

⟨p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & −(1 + s)e_{1} & 0 \\ u_{3} & u_{2} & −u_{4} & 0 & (1 + s)e_{1} & 0 & (s − 1)e_{2} \\ u_{4} & 0 & u_{1} & 0 & 0 & (1 − s)e_{2} & 0 \end{matrix}

s ⩾ 0

6.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & −x & 0 & 0 & 0 \\ 0 & 0 & 0 & x & 0 \end{matrix} )| x ∈ ℝ\} ⋌ 𝔫_{5}, 𝔤 = ⟨p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & e_{2} & 0 \\ u_{3} & u_{2} & −u_{4} & 0 & −e_{2} & 0 & e_{1} \\ u_{4} & 0 & u_{1} & 0 & 0 & −e_{1} & 0 \end{matrix}

7. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 2.5^{2}

\begin{matrix}[ ,] & e_{1} & e_{2} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{2} & −u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{1} & 0 & 0 & 0 & 0 \end{matrix}

3.1^{1}

𝔤 = \{( \begin{matrix}x & z & 0 & 0 \\ 0 & y & 0 & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −z & −y \end{matrix} )| x, y, z ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.1^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & e_{3} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & −e_{3} & 0 & u_{2} & 0 & −u_{4} \\ e_{3} & −e_{3} & e_{3} & 0 & 0 & u_{1} & −u_{4} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{2} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{4} & 0 & 0 & 0 & 0 & 0 \end{matrix}

EINSTEIN–MAXWELL EQUATION 97

3.1^{2}

𝔤 = \{( \begin{matrix}x & −y & z & 0 \\ y & x & 0 & z \\ 0 & 0 & −x & −y \\ 0 & 0 & y & −x \end{matrix} )| x, y, z ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.1^{2}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{2} & 0 & 0 & 0 & u_{2} & −u_{1} & u_{4} & −u_{3} \\ e_{3} & −2e_{3} & 0 & 0 & 0 & 0 & u_{1} & u_{2} \\ u_{1} & −u_{1} & −u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & −u_{4} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{4} & u_{4} & u_{3} & −u_{2} & 0 & 0 & 0 & 0 \end{matrix}

3.2^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & z \\ 0 & λx & −z & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −y & −λx \end{matrix} )| x, y, z ∈ ℝ, λ ⩾ 0\}

λ = 0

1.

𝔤̄ = \{( \begin{matrix}t & 0 & 0 \\ 0 & t + x & y \\ 0 & z & −x \end{matrix} )| x, y, z, t ∈ ℝ\} ⋌ 𝔫_{3},

𝔤 = ⟨( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) + h, p⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & e_{3} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & −2e_{2} \\ e_{3} & −e_{3} & 0 & 0 & 0 & −2e_{3} & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 2e_{3}−u_{1} & u_{2}+u_{4} & 2e_{2}−u_{1} \\ u_{2} & 0 & −u_{1} & 2e_{3} & u_{1}−2e_{3} & 0 & −2u_{3} & u_{2}−u_{4} \\ u_{3} & u_{3} & u_{4} & u_{2} & −u_{2}−u_{4} & 2u_{3} & 0 & 2u_{3} \\ u_{4} & 0 & 2e_{2} & −u_{1} & u_{1}−2e_{2} & u_{4}−u_{2} & −2u_{3} & 0 \end{matrix}

98 B. KOMRAKOV, JNR.

2.

𝔤̄ = \{( \begin{matrix}t & 0 & 0 \\ 0 & t + x & y \\ 0 & z & −x \end{matrix} )| x, y, z, t ∈ ℝ\} ⋌ ℝ^{3},

𝔤 = \{( \begin{matrix}t & 0 & 0 \\ 0 & t & y \\ 0 & 0 & 0 \end{matrix} ) ⋌ ( \begin{matrix}u \\ −u \\ 0 \end{matrix} )| y, t, u ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & e_{3} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & −2e_{2} \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & u_{2} & −u_{1} \\ u_{2} & 0 & −u_{1} & 0 & 0 & 0 & 0 & u_{2} \\ u_{3} & u_{3} & u_{4} & u_{2} & −u_{2} & 0 & 0 & 2u_{3} \\ u_{4} & 0 & 2e_{2} & −u_{1} & u_{1} & −u_{2} & −2u_{3} & 0 \end{matrix}

λ = 1 3.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −y & 0 \\ 0 & 0 & 2x & 0 & 0 \\ 0 & 0 & y & x & 0 \\ 0 & y & 0 & 0 & −x \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & −1 \end{matrix} ) , p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & −2e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & e_{2} & 0 \\ u_{3} & u_{3} & u_{4} & u_{2} & 0 & −e_{2} & 0 & 0 \\ u_{4} & u_{4} & 0 & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

EINSTEIN–MAXWELL EQUATION 99

λ ⩾ 0

4. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.2^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & (1−λ)e_{2} & (1+λ)e_{3} & u_{1} & λu_{2} & −u_{3} & −λu_{4} \\ e_{2} & (λ−1)e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & −(1+λ)e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −λu_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & λu_{4} & 0 & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

3.2^{2}

𝔤 = \{( \begin{matrix}x & y & 0 & −z \\ 0 & 0 & −y & −λx \\ 0 & 0 & −x & 0 \\ 0 & λx & z & 0 \end{matrix} )| x, y, z ∈ ℝ, λ ⩾ 0\}

λ = 0

1.

𝔤̄ = \{( \begin{matrix}t & 0 & 0 \\ 0 & t + x & y \\ 0 & z & −x \end{matrix} )| x, y, z, t ∈ ℝ\} ⋌ 𝔫_{3},

𝔤 = ⟨( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) − h, p⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} & e_{3} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & −e_{2} & 0 & 0 & 0 & −e_{2} + u_{1} & −u_{2} & e_{3} \\ e_{3} & −e_{3} & 0 & 0 & 0 & −e_{3} & u_{4} & −e_{2} − u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & e_{2} − u_{1} & u_{2} & −e_{3} \\ u_{2} & 0 & e_{2} − u_{1} & e_{3} & −e_{2} + u_{1} & 0 & −2u_{3} & −u_{4} \\ u_{3} & u_{3} & u_{2} & −u_{4} & −u_{2} & 2u_{3} & 0 & 0 \\ u_{4} & 0 & −e_{3} & e_{2} + u_{1} & e_{3} & u_{4} & 0 & 0 \end{matrix}

100 B. KOMRAKOV, JNR.

λ ⩾ 0 2.

𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.2^{2}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & e_{2} − λe_{3} & e_{3} + λe_{2} & u_{1} & λu_{4} & −u_{3} & −λu_{2} \\ e_{2} & λe_{3} − e_{2} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{3} & −e_{3} − λe_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −λu_{4} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{2} & −u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & λu_{2} & 0 & u_{1} & 0 & 0 & 0 & 0 \end{matrix}

3.3^{1}

𝔤 = \{( \begin{matrix}0 & y & 0 & z \\ 0 & x & −z & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −y & −x \end{matrix} )| x, y, z ∈ ℝ\}

1.

𝔤̄ = \{( \begin{matrix}y & 0 & 0 & 0 & 0 \\ 0 & −x & 0 & 0 & py \\ 0 & 0 & x & py & 0 \\ 0 & 0 & −y & x + y & 0 \\ 0 & −y & 0 & 0 & y − x \end{matrix} )| x, y ∈ ℝ, p ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & −1 \end{matrix} ) , p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{2} & e_{3} & 0 & u_{2} & 0 & −u_{4} \\ e_{2} & e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & pe_{3}+u_{2} & 0 \\ u_{3} & 0 & u_{4} & u_{2} & −u_{1} & −pe_{3}−u_{2} & 0 & −pe_{2}−u_{4} \\ u_{4} & u_{4} & 0 & −u_{1} & 0 & 0 & pe_{2}+u_{4} & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & −x & 0 & 0 & y \\ 0 & 0 & x & y & 0 \\ 0 & 0 & −y & x & 0 \\ 0 & −y & 0 & 0 & −x \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & −1 \end{matrix} ) , p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{2} & e_{3} & 0 & u_{2} & 0 & −u_{4} \\ e_{2} & e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & e_{3} & 0 \\ u_{3} & 0 & u_{4} & u_{2} & 0 & −e_{3} & 0 & −e_{2} \\ u_{4} & u_{4} & 0 & −u_{1} & 0 & 0 & e_{2} & 0 \end{matrix}

3.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & −x & 0 & 0 & −y \\ 0 & 0 & x & −y & 0 \\ 0 & 0 & −y & x & 0 \\ 0 & −y & 0 & 0 & −x \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & −1 \end{matrix} ) , p_{1}, p_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{2} & e_{3} & 0 & u_{2} & 0 & −u_{4} \\ e_{2} & e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & −e_{3} & 0 \\ u_{3} & 0 & u_{4} & u_{2} & 0 & e_{3} & 0 & e_{2} \\ u_{4} & u_{4} & 0 & −u_{1} & 0 & 0 & −e_{2} & 0 \end{matrix}

4. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.3^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{2} & e_{3} & 0 & u_{2} & 0 & −u_{4} \\ e_{2} & e_{2} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & 0 & u_{4} & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & u_{4} & 0 & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

3.3^{2}

𝔤 = \{( \begin{matrix}0 & y & 0 & −z \\ 0 & 0 & −y & −x \\ 0 & 0 & 0 & 0 \\ 0 & x & z & 0 \end{matrix} )| x, y, z ∈ ℝ\}

1.

𝔤̄ = \{( \begin{matrix}y & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & py & x \\ 0 & 0 & y & x & y \\ 0 & −y & −x & y & 0 \\ 0 & −x & −py & 0 & 0 \end{matrix} )| x, y ∈ ℝ, p ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \end{matrix} ) , p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & 0 & u_{4} & 0 & −u_{2} \\ e_{2} & e_{3} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{3} & −e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & u_{1} & 0 \\ u_{2} & −u_{4} & −u_{1} & 0 & 0 & 0 & pe_{2} + u_{2} & 0 \\ u_{3} & 0 & u_{2} & −u_{4} & −u_{1} & −pe_{2} − u_{2} & 0 & pe_{3} − u_{4} \\ u_{4} & u_{2} & 0 & u_{1} & 0 & 0 & −pe_{3} + u_{4} & 0 \end{matrix}

2.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & y & x \\ 0 & 0 & 0 & x & y \\ 0 & −y & −x & 0 & 0 \\ 0 & −x & −y & 0 & 0 \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \end{matrix} ) , p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & 0 & u_{4} & 0 & −u_{2} \\ e_{2} & e_{3} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{3} & −e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{4} & −u_{1} & 0 & 0 & 0 & e_{2} & 0 \\ u_{3} & 0 & u_{2} & −u_{4} & 0 & −e_{2} & 0 & e_{3} \\ u_{4} & u_{2} & 0 & u_{1} & 0 & 0 & −e_{3} & 0 \end{matrix}

3.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −y & x \\ 0 & 0 & 0 & x & y \\ 0 & −y & −x & 0 & 0 \\ 0 & −x & y & 0 & 0 \end{matrix} )| x, y ∈ ℝ\} ⋌ 𝔫_{5},

𝔤 = ⟨( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \end{matrix} ) , p_{1}, q_{2}⟩

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & 0 & u_{4} & 0 & −u_{2} \\ e_{2} & e_{3} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{3} & −e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{4} & −u_{1} & 0 & 0 & 0 & −e_{2} & 0 \\ u_{3} & 0 & u_{2} & −u_{4} & 0 & e_{2} & 0 & −e_{3} \\ u_{4} & u_{2} & 0 & u_{1} & 0 & 0 & e_{3} & 0 \end{matrix}

4. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.3^{2}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & 0 & u_{4} & 0 & −u_{2} \\ e_{2} & e_{3} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{3} & −e_{2} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{4} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & 0 & u_{2} & −u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & u_{2} & 0 & u_{1} & 0 & 0 & 0 & 0 \end{matrix}

3.4^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & 0 \\ z & −x & 0 & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & x \end{matrix} )| x, y, z ∈ ℝ\}

EINSTEIN–MAXWELL EQUATION 105

1. 𝔤̄ = 𝔰𝔩(2, ℝ) ⋌ (V ⊕ V^{∗}), where V = ℝ^{2} is natural 𝔰𝔩(2, ℝ)-module, 𝔤 = 𝔰𝔩(2, ℝ)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & u_{2} & 0 & 0 & −u_{3} \\ u_{1} & −u_{1} & 0 & −u_{2} & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & −u_{4} & 0 & u_{3} & 0 & 0 & 0 & 0 \end{matrix}

3.4^{2}

𝔤 = \{( \begin{matrix}0 & y & −x & −z \\ −y & 0 & −z & x \\ x & z & 0 & y \\ z & −x & −y & 0 \end{matrix} )| x, y, z ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 3.4^{2} ≅ 𝔰𝔲(2), 𝔤 acts irreducibly on ℝ^{4}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{3} & −2e_{2} & u_{3} & −u_{4} & −u_{1} & u_{2} \\ e_{2} & −2e_{3} & 0 & 2e_{1} & −u_{2} & u_{1} & −u_{4} & u_{3} \\ e_{3} & 2e_{2} & −2e_{1} & 0 & u_{4} & u_{3} & −u_{2} & −u_{1} \\ u_{1} & −u_{3} & u_{2} & −u_{4} & 0 & 0 & 0 & 0 \\ u_{2} & u_{4} & −u_{1} & −u_{3} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & u_{4} & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{2} & −u_{3} & u_{1} & 0 & 0 & 0 & 0 \end{matrix}

3.5^{1}

𝔤 = \{( \begin{matrix}2x & y & 0 & 0 \\ 2z & 0 & −2y & 0 \\ 0 & −z & −2x & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x, y, z ∈ ℝ\}

1. 𝔤̄ = (𝔰𝔬(2, 1) ⊕ ℝ id_{ℝ^{3}}) ⋌ ℝ^{3}, 𝔤 = 𝔰𝔬(2, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & 2u_{1} & 0 & −2u_{3} & 0 \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & u_{1} & −2u_{2} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & 2u_{2} & −u_{3} & 0 & 0 \\ u_{1} & −2u_{1} & 0 & −2u_{2} & 0 & 0 & 0 & u_{1} \\ u_{2} & 0 & −u_{1} & u_{3} & 0 & 0 & 0 & u_{2} \\ u_{3} & 2u_{3} & 2u_{2} & 0 & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & 0 & 0 & −u_{1} & −u_{2} & −u_{3} & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔬(2, 2) × ℝ, 𝔤 = 𝔰𝔬(2, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & 2u_{1} & 0 & −2u_{3} & 0 \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & u_{1} & −2u_{2} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & 2u_{2} & −u_{3} & 0 & 0 \\ u_{1} & −2u_{1} & 0 & −2u_{2} & 0 & e_{2} & e_{1} & 0 \\ u_{2} & 0 & −u_{1} & u_{3} & −e_{2} & 0 & e_{3} & 0 \\ u_{3} & 2u_{3} & 2u_{2} & 0 & −e_{1} & −e_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

3. 𝔤̄ = 𝔰𝔬(3, 1) × ℝ, 𝔤 = 𝔰𝔬(2, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & 2u_{1} & 0 & −2u_{3} & 0 \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & u_{1} & −2u_{2} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & 2u_{2} & −u_{3} & 0 & 0 \\ u_{1} & −2u_{1} & 0 & −2u_{2} & 0 & −e_{2} & −e_{1} & 0 \\ u_{2} & 0 & −u_{1} & u_{3} & e_{2} & 0 & −e_{3} & 0 \\ u_{3} & 2u_{3} & 2u_{2} & 0 & e_{1} & e_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

4. 𝔤̄ = (𝔰𝔬(2, 1) ⋌ ℝ^{3}) × ℝ, 𝔤 = 𝔰𝔬(2, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & 2u_{1} & 0 & −2u_{3} & 0 \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & u_{1} & −2u_{2} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & 2u_{2} & −u_{3} & 0 & 0 \\ u_{1} & −2u_{1} & 0 & −2u_{2} & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{1} & u_{3} & 0 & 0 & 0 & 0 \\ u_{3} & 2u_{3} & 2u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

3.5^{2}

𝔤 = \{( \begin{matrix}0 & x & y & 0 \\ −x & 0 & z & 0 \\ −y & −z & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )| x, y, z ∈ ℝ\}

1. 𝔤̄ = (𝔰𝔬(3) ⊕ ℝ id_{ℝ^{3}}) ⋌ ℝ^{3}, 𝔤 = 𝔰𝔬(3)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{3} & 0 & −e_{1} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & −e_{2} & e_{1} & 0 & 0 & −u_{3} & u_{2} & 0 \\ u_{1} & u_{2} & u_{3} & 0 & 0 & 0 & 0 & u_{1} \\ u_{2} & −u_{1} & 0 & u_{3} & 0 & 0 & 0 & u_{2} \\ u_{3} & 0 & −u_{1} & −u_{2} & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & 0 & 0 & −u_{1} & −u_{2} & −u_{3} & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔬(3, 1) × ℝ, 𝔤 = 𝔰𝔬(3)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{3} & 0 & −e_{1} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & −e_{2} & e_{1} & 0 & 0 & −u_{3} & u_{2} & 0 \\ u_{1} & u_{2} & u_{3} & 0 & 0 & e_{1} & e_{2} & 0 \\ u_{2} & −u_{1} & 0 & u_{3} & −e_{1} & 0 & e_{3} & 0 \\ u_{3} & 0 & −u_{1} & −u_{2} & −e_{2} & −e_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

3. 𝔤̄ = 𝔰𝔬(4) × ℝ, 𝔤 = 𝔰𝔬(3)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{3} & 0 & −e_{1} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & −e_{2} & e_{1} & 0 & 0 & −u_{3} & u_{2} & 0 \\ u_{1} & u_{2} & u_{3} & 0 & 0 & −e_{1} & −e_{2} & 0 \\ u_{2} & −u_{1} & 0 & u_{3} & e_{1} & 0 & −e_{3} & 0 \\ u_{3} & 0 & −u_{1} & −u_{2} & e_{2} & e_{3} & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

4. 𝔤̄ = (𝔰𝔬(3) ⋌ ℝ^{3}) × ℝ, 𝔤 = 𝔰𝔬(3)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{3} & e_{2} & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{3} & 0 & −e_{1} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & −e_{2} & e_{1} & 0 & 0 & −u_{3} & u_{2} & 0 \\ u_{1} & u_{2} & u_{3} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & u_{3} & 0 & 0 & 0 & 0 \\ u_{3} & 0 & −u_{1} & −u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

4.1^{1}

𝔤 = \{( \begin{matrix}x & z & 0 & t \\ 0 & y & −t & 0 \\ 0 & 0 & −x & 0 \\ 0 & 0 & −z & −y \end{matrix} )| x, y, z, t ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 4.1^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & e_{3} & e_{4} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & −e_{3} & e_{4} & 0 & u_{2} & 0 & −u_{4} \\ e_{3} & −e_{3} & e_{3} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & −e_{4} & −e_{4} & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & u_{4} & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{4} & 0 & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

4.1^{2}

𝔤 = \{( \begin{matrix}x & z & 0 & −t \\ 0 & 0 & −z & −y \\ 0 & 0 & −x & 0 \\ 0 & y & t & 0 \end{matrix} )| x, y, z, t ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & e_{3} & e_{4} & u_{1} & 0 & −u_{3} & 0 \\ e_{2} & 0 & 0 & −e_{4} & e_{3} & 0 & u_{4} & 0 & −u_{2} \\ e_{3} & −e_{3} & e_{4} & 0 & 0 & 0 & u_{1} & −u_{2} & 0 \\ e_{4} & −e_{4} & −e_{3} & 0 & 0 & 0 & 0 & u_{4} & −u_{1} \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & 0 & −u_{4} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & 0 & u_{2} & −u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & u_{2} & 0 & u_{1} & 0 & 0 & 0 & 0 \end{matrix}

4.2^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & 0 \\ z & t & 0 & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & −t \end{matrix} )| x, y, z, t ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔩(3, ℝ), 𝔤 = 𝔤𝔩(2, ℝ)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & u_{2} & 0 & 0 & −u_{3} \\ u_{1} & −u_{1} & −u_{1} & 0 & −u_{2} & 0 & 0 & e_{1} + 3e_{2} & 2e_{3} \\ u_{2} & u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & 2e_{4} & −e_{1} + 3e_{2} \\ u_{3} & u_{3} & u_{3} & u_{4} & 0 & −e_{1} − 3e_{2} & −2e_{4} & 0 & 0 \\ u_{4} & −u_{4} & u_{4} & 0 & u_{3} & −2e_{3} & e_{1} − 3e_{2} & 0 & 0 \end{matrix}

2. 𝔤̄ = 𝔤𝔩(2, ℝ) ⋌ (V ⊕ V^{∗}), where V = ℝ^{2} is natural 𝔤𝔩(2, ℝ)-module, 𝔤 = 𝔤𝔩(2, ℝ)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & u_{2} & 0 & 0 & −u_{3} \\ u_{1} & −u_{1} & −u_{1} & 0 & −u_{2} & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{3} & u_{4} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & −u_{4} & u_{4} & 0 & u_{3} & 0 & 0 & 0 & 0 \end{matrix}

4.2^{2}

𝔤 = \{( \begin{matrix}0 & y & −x & −z \\ −y & 0 & −z & −t \\ x & z & 0 & y \\ z & t & −y & 0 \end{matrix} )| x, y, z, t ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔲(3), 𝔤 = 𝔲(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{4} & −2e_{3} & u_{3} & −u_{4} & −u_{1} & u_{2} \\ e_{2} & 0 & 0 & 0 & 0 & u_{3} & u_{4} & −u_{1} & −u_{2} \\ e_{3} & −2e_{4} & 0 & 0 & 2e_{1} & −u_{2} & u_{1} & −u_{4} & u_{3} \\ e_{4} & 2e_{3} & 0 & −2e_{1} & 0 & u_{4} & u_{3} & −u_{2} & −u_{1} \\ u_{1} & −u_{3} & −u_{3} & u_{2} & −u_{4} & 0 & −e_{3} & e_{1} + 3e_{2} & e_{4} \\ u_{2} & u_{4} & −u_{4} & −u_{1} & −u_{3} & e_{3} & 0 & e_{4} & −e_{1} + 3e_{2} \\ u_{3} & u_{1} & u_{1} & u_{4} & u_{2} & −e_{1} − 3e_{2} & −e_{4} & 0 & −e_{3} \\ u_{4} & −u_{2} & u_{2} & −u_{3} & u_{1} & −e_{4} & e_{1} − 3e_{2} & e_{3} & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔲(2, 1), 𝔤 = 𝔲(2)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{4} & −2e_{3} & u_{3} & −u_{4} & −u_{1} & u_{2} \\ e_{2} & 0 & 0 & 0 & 0 & u_{3} & u_{4} & −u_{1} & −u_{2} \\ e_{3} & −2e_{4} & 0 & 0 & 2e_{1} & −u_{2} & u_{1} & −u_{4} & u_{3} \\ e_{4} & 2e_{3} & 0 & −2e_{1} & 0 & u_{4} & u_{3} & −u_{2} & −u_{1} \\ u_{1} & −u_{3} & −u_{3} & u_{2} & −u_{4} & 0 & e_{3} & −e_{1} − 3e_{2} & −e_{4} \\ u_{2} & u_{4} & −u_{4} & −u_{1} & −u_{3} & −e_{3} & 0 & −e_{4} & e_{1} − 3e_{2} \\ u_{3} & u_{1} & u_{1} & u_{4} & u_{2} & e_{1} + 3e_{2} & e_{4} & 0 & e_{3} \\ u_{4} & −u_{2} & u_{2} & −u_{3} & u_{1} & e_{4} & 3e_{2} − e_{1} & −e_{3} & 0 \end{matrix}

3. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 4.2^{2} ≅ 𝔲(2), 𝔤 acts irreducibly on ℝ^{4}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{4} & −2e_{3} & u_{3} & −u_{4} & −u_{1} & u_{2} \\ e_{2} & 0 & 0 & 0 & 0 & u_{3} & u_{4} & −u_{1} & −u_{2} \\ e_{3} & −2e_{4} & 0 & 0 & 2e_{1} & −u_{2} & u_{1} & −u_{4} & u_{3} \\ e_{4} & 2e_{3} & 0 & −2e_{1} & 0 & u_{4} & u_{3} & −u_{2} & −u_{1} \\ u_{1} & −u_{3} & −u_{3} & u_{2} & −u_{4} & 0 & 0 & 0 & 0 \\ u_{2} & u_{4} & −u_{4} & −u_{1} & −u_{3} & 0 & 0 & 0 & 0 \\ u_{3} & u_{1} & u_{1} & u_{4} & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{2} & u_{2} & −u_{3} & u_{1} & 0 & 0 & 0 & 0 \end{matrix}

4.2^{3}

𝔤 = \{( \begin{matrix}x & y & 0 & t \\ z & −x & −t & 0 \\ 0 & t & −x & −z \\ −t & 0 & −y & x \end{matrix} )| x, y, z, t ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔲(2, 1), 𝔤 = 𝔤𝔩(2, ℝ)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & −u_{4} & u_{3} & −u_{2} & u_{1} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & u_{2} & 0 & 0 & −u_{3} \\ u_{1} & −u_{1} & u_{4} & 0 & −u_{2} & 0 & 3e_{2} & e_{1} & 2e_{3} \\ u_{2} & u_{2} & −u_{3} & −u_{1} & 0 & −3e_{2} & 0 & 2e_{4} & −e_{1} \\ u_{3} & u_{3} & u_{2} & u_{4} & 0 & −e_{1} & −2e_{4} & 0 & 3e_{2} \\ u_{4} & −u_{4} & −u_{1} & 0 & u_{3} & −2e_{3} & e_{1} & −3e_{2} & 0 \end{matrix}

2. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 4.2^{3} ≅ 𝔤𝔩(2, ℝ), 𝔤 acts irreducibly on ℝ^{4}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & −u_{4} & u_{3} & −u_{2} & u_{1} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & u_{2} & 0 & 0 & −u_{3} \\ u_{1} & −u_{1} & u_{4} & 0 & −u_{2} & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{3} & −u_{1} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{2} & u_{4} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & −u_{4} & −u_{1} & 0 & u_{3} & 0 & 0 & 0 & 0 \end{matrix}

4.3^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & t \\ z & −x & −t & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & x \end{matrix} )| x, y, z, t ∈ ℝ\}

1.

𝔤̄ = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & x & y & 0 & 0 \\ 0 & z & −x & 0 & 0 \\ 0 & 0 & 0 & x & y \\ 0 & 0 & 0 & z & −x \end{matrix} )| x, y, z ∈ ℝ\} ⋌ 𝔫̃_{5},

𝔤 = \{( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & x & y & 0 & 0 \\ 0 & z & −x & 0 & 0 \\ 0 & 0 & 0 & x & y \\ 0 & 0 & 0 & z & −x \end{matrix} )| x, y, z ∈ ℝ\} × ℝh

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & 0 & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & 0 & u_{2} & 0 & 0 & −u_{3} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & −u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & 0 & u_{2} & 0 & 0 & 0 & e_{4} \\ u_{4} & −u_{4} & 0 & u_{3} & −u_{1} & 0 & 0 & −e_{4} & 0 \end{matrix}

2. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 4.3^{1} ≅ 𝔤𝔩(2, ℝ)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & 0 & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{3} & 2e_{3} & −e_{1} & 0 & 0 & u_{2} & 0 & 0 & −u_{3} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & 0 & −u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{4} & 0 & u_{3} & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

5.1^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & u \\ z & t & −u & 0 \\ 0 & 0 & −x & −z \\ 0 & 0 & −y & −t \end{matrix} )| x, y, z, t, u ∈ ℝ\}

1. 𝔤̄ = 𝔤 ⋌ ℝ^{4}, 𝔤 = 5.1^{1}

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & 0 & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & 2e_{5} & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & 0 & u_{2} & 0 & 0 & −u_{3} \\ e_{5} & 0 & −2e_{5} & 0 & 0 & 0 & 0 & 0 & −u_{2} & u_{1} \\ u_{1} & −u_{1} & −u_{1} & 0 & −u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{2} & −u_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{3} & u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 \\ u_{4} & −u_{4} & u_{4} & 0 & u_{3} & −u_{1} & 0 & 0 & 0 & 0 \end{matrix}

6.1^{1}

𝔤 = \{( \begin{matrix}x & y & 0 & u \\ z & t & −u & 0 \\ 0 & v & −x & −z \\ −v & 0 & −y & −t \end{matrix} )| x, y, z, t, u, v ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔬(3, 2), 𝔤 = 𝔰𝔬(2, 2)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & 0 & 0 & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & 2e_{5} & −2e_{6} & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & 0 & 0 & u_{2} & 0 & 0 & −u_{3} \\ e_{5} & 0 & −2e_{5} & 0 & 0 & 0 & −e_{2} & 0 & 0 & −u_{2} & u_{1} \\ e_{6} & 0 & 2e_{6} & 0 & 0 & e_{2} & 0 & −u_{4} & u_{3} & 0 & 0 \\ u_{1} & −u_{1} & −u_{1} & 0 & −u_{2} & 0 & u_{4} & 0 & 2e_{5} & e_{1} + e_{2} & 2e_{3} \\ u_{2} & u_{2} & −u_{2} & −u_{1} & 0 & 0 & −u_{3} & −2e_{5} & 0 & 2e_{4} & −e_{1} + e_{2} \\ u_{3} & u_{3} & u_{3} & u_{4} & 0 & u_{2} & 0 & −e_{1} − e_{2} & −2e_{4} & 0 & 2e_{6} \\ u_{4} & −u_{4} & u_{4} & 0 & u_{3} & −u_{1} & 0 & −2e_{3} & e_{1} − e_{2} & −2e_{6} & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔬(2, 2) ⋌ ℝ^{4}, 𝔤 = 𝔰𝔬(2, 2)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & 0 & 2e_{3} & −2e_{4} & 0 & 0 & u_{1} & −u_{2} & −u_{3} & u_{4} \\ e_{2} & 0 & 0 & 0 & 0 & 2e_{5} & −2e_{6} & u_{1} & u_{2} & −u_{3} & −u_{4} \\ e_{3} & −2e_{3} & 0 & 0 & e_{1} & 0 & 0 & 0 & u_{1} & −u_{4} & 0 \\ e_{4} & 2e_{4} & 0 & −e_{1} & 0 & 0 & 0 & u_{2} & 0 & 0 & −u_{3} \\ e_{5} & 0 & −2e_{5} & 0 & 0 & 0 & −e_{2} & 0 & 0 & −u_{2} & u_{1} \\ e_{6} & 0 & 2e_{6} & 0 & 0 & e_{2} & 0 & −u_{4} & u_{3} & 0 & 0 \\ u_{1} & −u_{1} & −u_{1} & 0 & −u_{2} & 0 & u_{4} & 0 & 0 & 0 & 0 \\ u_{2} & u_{2} & −u_{2} & −u_{1} & 0 & 0 & −u_{3} & 0 & 0 & 0 & 0 \\ u_{3} & u_{3} & u_{3} & u_{4} & 0 & u_{2} & 0 & 0 & 0 & 0 & 0 \\ u_{4} & −u_{4} & u_{4} & 0 & u_{3} & −u_{1} & 0 & 0 & 0 & 0 & 0 \end{matrix}

6.1^{2}

𝔤 = \{( \begin{matrix}0 & x & y & z \\ −x & 0 & t & u \\ −y & −t & 0 & v \\ −z & −u & −v & 0 \end{matrix} )| x, y, z, t, u, v ∈ ℝ\}

1.

𝔤̄ = 𝔰𝔬(4, 1), 𝔤 = 𝔰𝔬(4)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{4} & −e_{5} & e_{2} & e_{3} & 0 & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{4} & 0 & −e_{6} & −e_{1} & 0 & e_{3} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & e_{5} & e_{6} & 0 & 0 & −e_{1} & −e_{2} & −u_{4} & 0 & 0 & u_{1} \\ e_{4} & −e_{2} & e_{1} & 0 & 0 & −e_{6} & e_{5} & 0 & −u_{3} & u_{2} & 0 \\ e_{5} & −e_{3} & 0 & e_{1} & e_{6} & 0 & −e_{4} & 0 & −u_{4} & 0 & u_{2} \\ e_{6} & 0 & −e_{3} & e_{2} & −e_{5} & e_{4} & 0 & 0 & 0 & −u_{4} & u_{3} \\ u_{1} & u_{2} & u_{3} & u_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & e_{3} \\ u_{2} & −u_{1} & 0 & 0 & u_{3} & u_{4} & 0 & −e_{1} & 0 & e_{4} & e_{5} \\ u_{3} & 0 & −u_{1} & 0 & −u_{2} & 0 & u_{4} & −e_{2} & −e_{4} & 0 & e_{6} \\ u_{4} & 0 & 0 & −u_{1} & 0 & −u_{2} & −u_{3} & −e_{3} & −e_{5} & −e_{6} & 0 \end{matrix}

2.

𝔤̄ = 𝔰𝔬(5), 𝔤 = 𝔰𝔬(4)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{4} & −e_{5} & e_{2} & e_{3} & 0 & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{4} & 0 & −e_{6} & −e_{1} & 0 & e_{3} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & e_{5} & e_{6} & 0 & 0 & −e_{1} & −e_{2} & −u_{4} & 0 & 0 & u_{1} \\ e_{4} & −e_{2} & e_{1} & 0 & 0 & −e_{6} & e_{5} & 0 & −u_{3} & u_{2} & 0 \\ e_{5} & −e_{3} & 0 & e_{1} & e_{6} & 0 & −e_{4} & 0 & −u_{4} & 0 & u_{2} \\ e_{6} & 0 & −e_{3} & e_{2} & −e_{5} & e_{4} & 0 & 0 & 0 & −u_{4} & u_{3} \\ u_{1} & u_{2} & u_{3} & u_{4} & 0 & 0 & 0 & 0 & −e_{1} & −e_{2} & −e_{3} \\ u_{2} & −u_{1} & 0 & 0 & u_{3} & u_{4} & 0 & e_{1} & 0 & −e_{4} & −e_{5} \\ u_{3} & 0 & −u_{1} & 0 & −u_{2} & 0 & u_{4} & e_{2} & e_{4} & 0 & −e_{6} \\ u_{4} & 0 & 0 & −u_{1} & 0 & −u_{2} & −u_{3} & e_{3} & e_{5} & e_{6} & 0 \end{matrix}

3.

𝔤̄ = 𝔰𝔬(4) ⋌ ℝ^{4}, 𝔤 = 𝔰𝔬(4)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{4} & −e_{5} & e_{2} & e_{3} & 0 & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{4} & 0 & −e_{6} & −e_{1} & 0 & e_{3} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & e_{5} & e_{6} & 0 & 0 & −e_{1} & −e_{2} & −u_{4} & 0 & 0 & u_{1} \\ e_{4} & −e_{2} & e_{1} & 0 & 0 & −e_{6} & e_{5} & 0 & −u_{3} & u_{2} & 0 \\ e_{5} & −e_{3} & 0 & e_{1} & e_{6} & 0 & −e_{4} & 0 & −u_{4} & 0 & u_{2} \\ e_{6} & 0 & −e_{3} & e_{2} & −e_{5} & e_{4} & 0 & 0 & 0 & −u_{4} & u_{3} \\ u_{1} & u_{2} & u_{3} & u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & u_{3} & u_{4} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & 0 & −u_{1} & 0 & −u_{2} & 0 & u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{1} & 0 & −u_{2} & −u_{3} & 0 & 0 & 0 & 0 \end{matrix}

6.1^{3}

𝔤 = \{( \begin{matrix}0 & x & y & z \\ −x & 0 & t & u \\ −y & −t & 0 & v \\ z & u & v & 0 \end{matrix} )| x, y, z, t, u, v ∈ ℝ\}

1. 𝔤̄ = 𝔰𝔬(3, 2), 𝔤 = 𝔰𝔬(3, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{4} & −e_{5} & e_{2} & e_{3} & 0 & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{4} & 0 & −e_{6} & −e_{1} & 0 & e_{3} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & e_{5} & e_{6} & 0 & 0 & e_{1} & e_{2} & u_{4} & 0 & 0 & u_{1} \\ e_{4} & −e_{2} & e_{1} & 0 & 0 & −e_{6} & e_{5} & 0 & −u_{3} & u_{2} & 0 \\ e_{5} & −e_{3} & 0 & −e_{1} & e_{6} & 0 & e_{4} & 0 & u_{4} & 0 & u_{2} \\ e_{6} & 0 & −e_{3} & −e_{2} & −e_{5} & −e_{4} & 0 & 0 & 0 & u_{4} & u_{3} \\ u_{1} & u_{2} & u_{3} & −u_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & −e_{3} \\ u_{2} & −u_{1} & 0 & 0 & u_{3} & −u_{4} & 0 & −e_{1} & 0 & e_{4} & −e_{5} \\ u_{3} & 0 & −u_{1} & 0 & −u_{2} & 0 & −u_{4} & −e_{2} & e_{4} & 0 & −e_{6} \\ u_{4} & 0 & 0 & −u_{1} & 0 & −u_{2} & −u_{3} & e_{3} & e_{5} & e_{6} & 0 \end{matrix}

2. 𝔤̄ = 𝔰𝔬(4, 1), 𝔤 = 𝔰𝔬(3, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{4} & −e_{5} & e_{2} & e_{3} & 0 & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{4} & 0 & −e_{6} & −e_{1} & 0 & e_{3} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & e_{5} & e_{6} & 0 & 0 & e_{1} & e_{2} & u_{4} & 0 & 0 & u_{1} \\ e_{4} & −e_{2} & e_{1} & 0 & 0 & −e_{6} & e_{5} & 0 & −u_{3} & u_{2} & 0 \\ e_{5} & −e_{3} & 0 & −e_{1} & e_{6} & 0 & e_{4} & 0 & u_{4} & 0 & u_{2} \\ e_{6} & 0 & −e_{3} & −e_{2} & −e_{5} & −e_{4} & 0 & 0 & 0 & u_{4} & u_{3} \\ u_{1} & u_{2} & u_{3} & −u_{4} & 0 & 0 & 0 & 0 & −e_{1} & −e_{2} & e_{3} \\ u_{2} & −u_{1} & 0 & 0 & u_{3} & −u_{4} & 0 & e_{1} & 0 & −e_{4} & e_{5} \\ u_{3} & 0 & −u_{1} & 0 & −u_{2} & 0 & −u_{4} & e_{2} & −e_{4} & 0 & e_{6} \\ u_{4} & 0 & 0 & −u_{1} & 0 & −u_{2} & −u_{3} & −e_{3} & −e_{5} & −e_{6} & 0 \end{matrix}

3. 𝔤̄ = 𝔰𝔬(3, 1) ⋌ ℝ^{4}, 𝔤 = 𝔰𝔬(3, 1)

\begin{matrix}[ ,] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & −e_{4} & −e_{5} & e_{2} & e_{3} & 0 & −u_{2} & u_{1} & 0 & 0 \\ e_{2} & e_{4} & 0 & −e_{6} & −e_{1} & 0 & e_{3} & −u_{3} & 0 & u_{1} & 0 \\ e_{3} & e_{5} & e_{6} & 0 & 0 & e_{1} & e_{2} & u_{4} & 0 & 0 & u_{1} \\ e_{4} & −e_{2} & e_{1} & 0 & 0 & −e_{6} & e_{5} & 0 & −u_{3} & u_{2} & 0 \\ e_{5} & −e_{3} & 0 & −e_{1} & e_{6} & 0 & e_{4} & 0 & u_{4} & 0 & u_{2} \\ e_{6} & 0 & −e_{3} & −e_{2} & −e_{5} & −e_{4} & 0 & 0 & 0 & u_{4} & u_{3} \\ u_{1} & u_{2} & u_{3} & −u_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2} & −u_{1} & 0 & 0 & u_{3} & −u_{4} & 0 & 0 & 0 & 0 & 0 \\ u_{3} & 0 & −u_{1} & 0 & −u_{2} & 0 & −u_{4} & 0 & 0 & 0 & 0 \\ u_{4} & 0 & 0 & −u_{1} & 0 & −u_{2} & −u_{3} & 0 & 0 & 0 & 0 \end{matrix}

CHAPTER II EINSTEIN–MAXWELL EQUATION ON

FOUR-DIMENSIONAL HOMOGENEOUS SPACES

Preliminaries:

Let (\overline{G}, M) be a homogeneous space with an invariant pseudo-Riemannian metric g. Let ω be an invariant differential 2-form on M which is closed (dω = 0) and coclosed (d(∗ω) = 0). The Einstein–Maxwell equation on the homogeneous space (\overline{G}, M) has the form:

r − λg = m_{ω},  (1)

where λ is an arbitrary real number, r is the Ricci tensor of g, m_{ω} is an invariant smooth tensor field of type (0,2) on M defined by m_{ω} = ω ∘ g^{−1} ∘ ω, where g, ω, m_{ω} are identified with the corresponding mappings T M → T^{∗}M.

Equation (1) is called the Einstein equation if m_{ω} ≡ 0.

Let (𝔤̄, 𝔤) be a pair of Lie algebras corresponding to the homogeneous space (\overline{G}, M). By 𝔪 denote the factor space 𝔤̄/𝔤 endowed with the natural structure of 𝔤-module, and let x_{𝔪} ∈ 𝔪 denote the image of the element x ∈ 𝔤 under the natural projection π: 𝔤̄ → 𝔪. Then, in terms of the pair (𝔤̄, 𝔤), the Einstein–Maxwell equation on the homogeneous space (\overline{G}, M) will have the form:

R − λB = M_{Ω},

where B is an invariant nondegenerate symmetric bilinear form on 𝔪, Ω is an invariant skew-symmetric bilinear form on 𝔪 which is closed (dΩ = 0) and coclosed (d(∗Ω) = 0), R is the invariant symmetric bilinear form on 𝔪 corresponding to the Ricci tensor, λ is an arbitrary real number, and M_{Ω} is the invariant symmetric bilinear form on 𝔪 corresponding to m_{ω}.

Now we identify 𝔪 with some (not necessarily 𝔤-invariant) subspace of 𝔤̄ complementary to 𝔤. Let E = {e_{1}, ..., e_{n}, u_{1}, u_{2}, u_{3}, u_{4}} be a basis of 𝔤̄, where {e_{1}, ..., e_{n}} is a basis of 𝔤 and U = {u_{1}, u_{2}, u_{3}, u_{4}} is a basis of 𝔪. Then we can identify any bilinear form B̃ on 𝔪 with the matrix of B̃ with respect to the basis U. We divide the solution of the Einstein–Maxwell equation on the pair (𝔤̄, 𝔤) into the following parts:

1. We find B and Ω. The pair (𝔤̄, 𝔤) uniquely defines the isotropic representation

ρ: 𝔤 → 𝔤𝔩(𝔪), ρ(x)(m) = [x, m]_{𝔪} for all x ∈ 𝔤, m ∈ 𝔪.

Let B̃ ∈ Bil(𝔪) be an arbitrary bilinear form on 𝔪. Then the invariance condition for B̃ has the form: ^{t}XB̃ + B̃X = 0 for all X ∈ ρ(𝔤). Verifying symmetry (skew-symmetry) condition, we obtain B(Ω).

2. Ω is closed, i.e., dΩ = 0. It means that ([F])

dΩ(x, y, z) = −Ω([x, y]_{𝔪}, z)+Ω([x, z]_{𝔪}, y)−Ω([y, z]_{𝔪}, x) = 0 for all x, y, z ∈ 𝔪.

3. Ω is coclosed, i.e., (. ∗ Ω) = 0, where ∗ is the Hodge operator corresponding to B.

4. We compute M_{Ω} = ΩB^{−1}Ω.

5. There is a one-to-one correspondence between the invariant linear connections on (\overline{G}, M) and the homomorphisms of 𝔤-modules Λ: 𝔤̄ → 𝔤𝔩(𝔪) such that Λ(x)(y_{𝔪}) = [x, y]_{𝔪} for all x ∈ 𝔤, y ∈ 𝔤̄. It follows that Λ(x) = ρ(x) for all x ∈ 𝔤. The invariant nondegenerate symmetric bilinear form B uniquely defines the invariant linear Levi–Civita connection Λ: 𝔤̄ → 𝔤𝔩(𝔪) in the following way. Let v: 𝔤̄×𝔤̄ → 𝔪 be the 𝔤-invariant symmetric mapping uniquely determined by the equality

2B(v(x, y), z_{𝔪}) = B(x_{𝔪}, [z, y]_{𝔪}) + B([z, x]_{𝔪}, y_{𝔪}) for all x, y, z ∈ 𝔤̄.

Then Λ(x)(y_{𝔪}) = \frac{1}{2}[x, y]_{𝔪} + v(x, y) for all x, y ∈ 𝔤̄.

6. To the curvature tensor of the connection Λ we assign the mapping K: 𝔪 × 𝔪 → 𝔤𝔩(𝔪) such that K(x, y) = [Λ(x), Λ(y)] − Λ([x, y]) for all x, y ∈ 𝔪.

7. To the Ricci tensor of the connection Λ we assign the symmetric bilinear form R: 𝔪 × 𝔪 → ℝ such that

R(u_{i}, u_{j}) = ∑_{l=1}^{4} K_{li}(u_{l}, u_{j}), u_{i}, u_{j} ∈ U, i, j = \overline{1, 4}.

8. We find the solutions. The solution of the Einstein–Maxwell equation is any triple (B, Ω, λ) satisfying the equality R − λB = M_{Ω}.

Remarks.

1. Note that all expressions are linear in all arguments. Therefore, in order to ensure that these conditions are satisfied for all arguments, we must only check that they hold for basis vectors.

2. By a, b, c, etc. we denote the entries of B, and by α, β, γ, etc. the entries of Ω. Unless otherwise stated, we assume that the entries of these matrices run through ℝ. Any additional conditions appear just after the matrices.

3. The matrix B must satisfy the condition det B ≠ 0.

4. For more details about parts 5–7, see [K-N].

Example. Consider the pair

1.1^{1}.2.

𝔤̄ = \{( \begin{matrix}x & 0 & 0 \\ 0 & y − x & 0 \\ 0 & 0 & py \end{matrix} )| x, y ∈ ℝ, p ∈ ℝ\}⋌ℝ^{3}, 𝔤 = \{( \begin{matrix}x & 0 & 0 \\ 0 & −x & 0 \\ 0 & 0 & 0 \end{matrix} )| x ∈ ℝ\}

\begin{matrix}[ ,] & e_{1} & u_{1} & u_{2} & u_{3} & u_{4} \\ e_{1} & 0 & u_{1} & 0 & −u_{3} & 0 \\ u_{1} & −u_{1} & 0 & 0 & 0 & 0 \\ u_{2} & 0 & 0 & 0 & 0 & pu_{2} \\ u_{3} & u_{3} & 0 & 0 & 0 & u_{3} \\ u_{4} & 0 & 0 & −pu_{2} & −u_{3} & 0 \end{matrix}

1. Checking invariance condition for arbitrary bilinear form B̃ on 𝔪 we obtain that

B̃ = ( \begin{matrix}0 & 0 & b_{1} & 0 \\ 0 & b_{2} & 0 & b_{3} \\ b_{4} & 0 & 0 & 0 \\ 0 & b_{5} & 0 & b_{6} \end{matrix} ) . Therefore, B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) .

2. dΩ(u_{3}, u_{4}, u_{1}) = −Ω([u_{3}, u_{4}], u_{1}) + Ω([u_{3}, u_{1}], u_{4}) − Ω([u_{4}, u_{1}], u_{3}) = α = 0 ⇒

α = 0. dΩ(u_{i}, u_{j}, u_{k}) = 0, i, j, k = \overline{1, 4}. It follows that Ω is closed ⟺ α = 0.

3. ∗Ω = ( \begin{matrix}0 & 0 & −a^{2}β & 0 \\ 0 & 0 & 0 & (bd − c^{2})α \\ a^{2}β & 0 & 0 & 0 \\ 0 & −(bd − c^{2})α & 0 & 0 \end{matrix} ) .

By conditions d(∗Ω) = 0 and α = 0 we obtain that Ω is closed and coclosed ⟺ Ω = 0.

4. M_{Ω} = ΩB^{−1}Ω = 0.

5. The mapping v|_{𝔪}: 𝔪 × 𝔪 → 𝔪 has the form:

v(u_{i}, u_{j})

u_{1}  u_{2}  u_{3}  u_{4}
u_{1}  0  0  \frac{ac}{2(bd−c^{2})}u_{2} − \frac{ab}{2(bd−c^{2})}u_{4}  \frac{1}{2}u_{1}
u_{2}  0  \frac{pbc}{bd−c^{2}} u_{2} − \frac{pb^{2}}{bd−c^{2}} u_{4}  0  \frac{p(bd+c^{2})}{2(bd−c^{2})}u_{2} − \frac{pbc}{bd−c^{2}} u_{4}
u_{3} \frac{ac}{2(bd−c^{2})}u_{2} − \frac{ab}{2(bd−c^{2})}u_{4}  0  0  0
u_{4}  \frac{1}{2}u_{1}  \frac{p(bd+c^{2})}{2(bd−c^{2})}u_{2} − \frac{pbc}{bd−c^{2}} u_{4}  0  \frac{pdc}{bd−c^{2}} u_{2} − \frac{pc^{2}}{bd−c^{2}} u_{4}

It follows that the invariant linear Levi–Civita connection Λ has the form:

Λ(e_{1}) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ),

Λ(u_{1}) = ( \begin{matrix}0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & \frac{ac}{2(bd−c^{2})} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −\frac{ab}{2(bd−c^{2})} & 0 \end{matrix} ) , Λ(u_{2}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & \frac{pbc}{bd−c^{2}} & 0 & \frac{pbd}{bd−c^{2}} \\ 0 & 0 & 0 & 0 \\ 0 & −\frac{pb^{2}}{bd−c^{2}} & 0 & −\frac{pbc}{bd−c^{2}} \end{matrix} ) ,^{Λ(u_{3}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ \frac{ac}{2(bd−c^{2})} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} \\ −\frac{ab}{2(bd−c^{2})} & 0 & 0 & 0 \end{matrix} ) , Λ(u_{4}) = ( \begin{matrix}\frac{1}{2} & 0 & 0 & 0 \\ 0 & \frac{pc^{2}}{bd−c^{2}} & 0 & \frac{pdc}{bd−c^{2}} \\ 0 & 0 & −\frac{1}{2} & 0 \\ 0 & −\frac{pbc}{bd−c^{2}} & 0 & −\frac{pc^{2}}{bd−c^{2}} \end{matrix} ) .}

6. The curvature tensor of the invariant linear Levi–Civita connection Λ has the form:

K(u_{1}, u_{2}) = ( \begin{matrix}0 & −\frac{pb^{2}}{2(bd−c^{2})} & 0 & −\frac{pbc}{2(bd−c^{2})} \\ 0 & 0 & \frac{pab}{2(bd−c^{2})} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) ,

K(u_{1}, u_{3}) = ( \begin{matrix}−\frac{ab}{4(bd−c^{2})} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{ab}{4(bd−c^{2})} & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) ,

K(u_{1}, u_{4}) = ( \begin{matrix}0 & −\frac{pbc}{2(bd−c^{2})} & 0 & −\frac{pc^{2}}{2(bd−c^{2})} − \frac{1}{4} \\ 0 & 0 & \frac{(2p−1)ac}{4(bd−c^{2})} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{ab}{4(bd−c^{2})} & 0 \end{matrix} ) ,

K(u_{2}, u_{3}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ −\frac{pab}{2(bd−c^{2})} & 0 & 0 & 0 \\ 0 & \frac{pb^{2}}{2(bd−c^{2})} & 0 & \frac{pbc}{2(bd−c^{2})} \\ 0 & 0 & 0 & 0 \end{matrix} ) ,

K(u_{2}, u_{4}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & −\frac{p^{2}bc}{bd−c^{2}} & 0 & −\frac{p^{2}bd}{bd−c^{2}} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{p^{2}b^{2}}{bd−c^{2}} & 0 & \frac{p^{2}bc}{bd−c^{2}} \end{matrix} ),

K(u_{3}, u_{4}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ \frac{(2p−1)ac}{4(bd−c^{2})} & 0 & 0 & 0 \\ 0 & −\frac{pbc}{2(bd−c^{2})} & 0 & −\frac{pc^{2}}{2(bd−c^{2})} − \frac{1}{4} \\ \frac{ab}{4(bd−c^{2})} & 0 & 0 & 0 \end{matrix} ) .

7. The Ricci tensor of the invariant linear Levi–Civita connection Λ has the form:

R = ( \begin{matrix}0 & 0 & −\frac{ab(p+1)}{2(bd−c^{2})} & 0 \\ 0 & −\frac{p(p+1)b^{2}}{bd−c^{2}} & 0 & −\frac{p(p+1)bc}{bd−c^{2}} \\ −\frac{ab(p+1)}{2(bd−c^{2})} & 0 & 0 & 0 \\ 0 & −\frac{p(p+1)bc}{bd−c^{2}} & 0 & −\frac{(p^{2}bd+pc^{2})}{bd−c^{2}} − \frac{1}{2} \end{matrix} ) .

8. The Einstein–Maxwell equation has the form: R − λB = M_{Ω}.

( \begin{matrix}0 & 0 & −\frac{ab(p+1)}{2(bd−c^{2})} & 0 \\ 0 & −\frac{p(p+1)b^{2}}{bd−c^{2}} & 0 & −\frac{p(p+1)bc}{bd−c^{2}} \\ −\frac{ab(p+1)}{2(bd−c^{2})} & 0 & 0 & 0 \\ 0 & −\frac{p(p+1)bc}{bd−c^{2}} & 0 & −\frac{(p^{2}bd+pc^{2})}{bd−c^{2}} − \frac{1}{2} \end{matrix} )−λ ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) = 0.

It follows that any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.2 has the form:

p = \frac{1}{2}

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = 0, λ = −\frac{3b}{4(bd − c^{2})}.

For other values of p the Einstein–Maxwell equation has no solutions.

1. Solutions of Einstein–Maxwell equation on pseudo-Riemannian pairs of codimension 4

Proposition 1.1^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) ,

where a^{2} = c^{2} − bd,

Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{3b − 2α^{2}}{2a^{2}} .

Proposition 1.1^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.2 has the form:

p = \frac{1}{2}

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = 0, λ = −\frac{3b}{4(bd − c^{2})}.

For other values of p the Einstein–Maxwell equation has no solutions.

Proposition 1.1^{1}.3. The Einstein–Maxwell equation has no solutions on the pair 1.1^{1}.3.

Proposition 1.1^{1}.4. The Einstein–Maxwell equation has no solutions on the pair 1.1^{1}.4.

Proposition 1.1^{1}.5. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.5 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{bd − c^{2}} ,

where α^{2} = \frac{b−β^{2}}{bd−c^{2}} a^{2} − a.

Proposition 1.1^{1}.6. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.6 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{bd − c^{2}} ,

where α^{2} = \frac{b−β^{2}}{bd−c^{2}} a^{2}.

Proposition 1.1^{1}.7. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.7 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{bd − c^{2}} ,

where α^{2} = −\frac{β^{2}a^{2}}{bd−c^{2}} − a.

Proposition 1.1^{1}.8. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.8 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & b \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{b^{2}} + 3a.

Proposition 1.1^{1}.9. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.9 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & b \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{b^{2}} .

Proposition 1.1^{1}.10. Any solution of the Einstein–Maxwell equation on the pair 1.1^{1}.10 has the form:

λ = 0

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & c \\ a & 0 & 0 & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{bd − c^{2}} ,

where α^{2} = −\frac{β^{2}a^{2}}{bd−c^{2}} .

λ = 1

B = ( \begin{matrix}0 & A \\ A^{t} & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & C \\ −C^{t} & 0 \end{matrix} ) , λ ∈ ℝ,

where A ∈ GL(2, ℝ), C ∈ Mat(2, ℝ), such that −λA = CA^{−1}C.

λ ∈]0, 1[

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & b \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{b^{2}} .

Proposition 1.1^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) ,

where 4a^{2} = bd − c^{2},

Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & −2α \\ −α & 0 & 0 & 0 \\ 0 & 2α & 0 & 0 \end{matrix} ) , λ = \frac{2α^{2} − 3b}{2a^{2}}.

Proposition 1.1^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.2 has the form:

p = 1

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = 0, λ = −\frac{3b}{bd − c^{2}} .

For other values of p the Einstein–Maxwell equation has no solutions.

Proposition 1.1^{2}.3. The Einstein–Maxwell equation has no solutions on the pair 1.1^{2}.3.

Proposition 1.1^{2}.4. The Einstein–Maxwell equation has no solutions on the pair 1.1^{2}.4.

Proposition 1.1^{2}.5. The Einstein–Maxwell equation has no solutions on the pair 1.1^{2}.5.

Proposition 1.1^{2}.6. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.6 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{bd − c^{2}} ,

where α^{2} = \frac{β^{2}−b}{bd−c^{2}} a^{2} − a.

Proposition 1.1^{2}.7. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.7 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{bd − c^{2}} ,

where α^{2} = \frac{β^{2}−b}{bd−c^{2}} a^{2} + a.

Proposition 1.1^{2}.8. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.8 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{bd − c^{2}} ,

where α^{2} = \frac{β^{2}−b}{bd−c^{2}} a^{2}.

Proposition 1.1^{2}.9. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.9 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{bd − c^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{bd−c^{2}} − a.

Proposition 1.1^{2}.10. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.10 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{bd − c^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{bd−c^{2}} + a.

Proposition 1.1^{2}.11. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.11 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{b^{2}} + 6a.

Proposition 1.1^{2}.12. Any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.12 has the form:

λ = 0

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{bd − c^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{bd−c^{2}} .

λ = 1

B = ( \begin{matrix}a & b & 0 & d \\ b & c & −d & 0 \\ 0 & −d & a & b \\ d & 0 & b & c \end{matrix} ) , Ω = ( \begin{matrix}0 & γ & α & δ \\ −γ & 0 & δ & β \\ −α & −δ & 0 & γ \\ −δ & −β & −γ & 0 \end{matrix} ) , λ ∈ ℝ,

such that −λB = M_{Ω}, where M_{Ω} = ΩB^{−1}Ω.

λ ∈]0, 1[

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}a^{2}}{b^{2}} .

Proposition 1.1^{3}.1. Any solution of the Einstein–Maxwell equation on the pair 1.1^{3}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = 0.

Proposition 1.1^{4}.1. Any solution of the Einstein–Maxwell equation on the pair 1.1^{4}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \\ 0 & 0 & a & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 1.1^{5}.1. Any solution of the Einstein–Maxwell equation on the pair 1.1^{5}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & b \\ 0 & 0 & b & −a \\ a & b & 0 & 0 \\ b & −a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & β \\ 0 & 0 & β & −α \\ −α & −β & 0 & 0 \\ −β & α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2} − β^{2}}{b^{2} − a^{2}} ,

where a^{2} ≠ b^{2}, αβ = \frac{β^{2}−α^{2}}{b^{2}−a^{2}} ab, or

B = ( \begin{matrix}0 & 0 & a & εa \\ 0 & 0 & εa & −a \\ a & εa & 0 & 0 \\ εa & −a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & δα \\ 0 & 0 & δα & −α \\ −α & −δα & 0 & 0 \\ −δα & α & 0 & 0 \end{matrix} ) , λ = −\frac{δα^{2}}{εa^{2}} ,

where ε ∈ {−1, 1}, δ ∈ {−1, 1}.

Proposition 1.1^{6}.1. Any solution of the Einstein–Maxwell equation on the pair 1.1^{6}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & b \\ 0 & 0 & −b & a \\ a & −b & 0 & 0 \\ b & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & β \\ 0 & 0 & −β & α \\ −α & β & 0 & 0 \\ −β & −α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2} − β^{2}}{b^{2} − a^{2}} ,

where a^{2} ≠ b^{2}, αβ = \frac{β^{2}−α^{2}}{b^{2}−a^{2}} ab, or

B = ( \begin{matrix}0 & 0 & a & εa \\ 0 & 0 & −εa & a \\ a & −εa & 0 & 0 \\ εa & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & δα \\ 0 & 0 & −δα & α \\ −α & δα & 0 & 0 \\ −δα & −α & 0 & 0 \end{matrix} ) , λ = −\frac{δα^{2}}{εa^{2}} ,

where ε ∈ {−1, 1}, δ ∈ {−1, 1}.

Proposition 1.2^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 1.2^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & \frac{αb}{a} & α \\ −α & −\frac{αb}{a} & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & β & 0 \\ 0 & −β & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = 0.

Proposition 1.2^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 1.2^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & −α & 0 \\ 0 & α & 0 & \frac{αb}{a} \\ −α & 0 & −\frac{αb}{a} & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0.

Proposition 1.3^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & β & α \\ 0 & 0 & α & γ \\ −β & −α & 0 & 0 \\ −α & −γ & 0 & 0 \end{matrix} ) , λ = \frac{3c}{8a^{2}} + \frac{βγ − α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, bc = d^{2}, 2αd = γb + βc, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & β & 0 \\ 0 & 0 & 0 & 0 \\ −β & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = \frac{3c}{8a^{2}} ,

where 8β^{2}c = 15(bc − d^{2}). Proposition 1.3^{1}.2. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.2. Proposition 1.3^{1}.3. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.3. Proposition 1.3^{1}.4. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.4. Proposition 1.3^{1}.5. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.5 has the form:

λ = 0, μ = 2

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & β & 0 \\ 0 & 0 & 0 & β \\ −β & 0 & 0 & −\frac{β(b+c)}{2a} \\ 0 & −β & \frac{β(b+c)}{2a} & 0 \end{matrix} ) , λ = \frac{β^{2}}{a^{2}} ,

where β ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ = 0, μ = −2

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & β & 0 \\ 0 & 0 & 0 & −3β \\ −β & 0 & 0 & \frac{2a}{3β} + \frac{β(3b−c)}{2a} \\ 0 & 3β & −\frac{2a}{3β} − \frac{β(3b−c)}{2a} & 0 \end{matrix}) , λ = −\frac{3β^{2}}{a^{2}} ,

where β ≠ 0.

For other values of λ, μ the Einstein–Maxwell equation has no solutions. Proposition 1.3^{1}.6. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.6. Proposition 1.3^{1}.7. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.7. Proposition 1.3^{1}.8. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.8 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & \frac{a}{4γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0. Proposition 1.3^{1}.9. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.9 has the form:

λ = 1

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & β & 0 \\ 0 & 0 & 0 & γ \\ −β & 0 & 0 & −\frac{γb+βc}{2a} \\ 0 & −γ & \frac{γb+βc}{2a} & 0 \end{matrix} ) , λ = \frac{βγ}{a^{2}} ,

where β^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ ≠ 1

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{(1−λ)^{2}a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & \frac{(1−λ)^{2}a}{4γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

Proposition 1.3^{1}.10. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.10 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & \frac{a}{4γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0.

Proposition 1.3^{1}.11. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.11 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{a}{γ} − \frac{γb}{2a} \\ 0 & −γ & \frac{a}{γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0.

Proposition 1.3^{1}.12. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.12 has the form:

λ = 1, μ = 2

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & β & α \\ 0 & 0 & α & γ \\ −β & −α & 0 & \frac{2αd−γb−βc}{2a} \\ −α & −γ & −\frac{2αd−γb−βc}{2a} & 0 \end{matrix}) , λ= \frac{βγ − α^{2}}{a^{2}} ,

where α^{2} + β^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ = 3, μ = 2

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & β & 0 \\ 0 & 0 & 0 & γ \\ −β & 0 & 0 & −\frac{γb+βc}{2a} \\ 0 & −γ & \frac{γb+βc}{2a} & 0 \end{matrix} ) , λ = \frac{βγ}{a^{2}} ,

where β^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

136 B. KOMRAKOV, JNR.

μ = λ + 1, λ ≠ 1

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & γ \\ 0 & −α & 0 & \frac{2αd−γb}{2a} \\ −α & −γ & −\frac{2αd−γb}{2a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

μ = λ − 1, λ ≠ 3

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{γb}{2a} \\ 0 & −γ & \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

μ ∉ {λ − 1, λ + 1}

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & \frac{(1−(λ−μ)^{2})a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & −\frac{(1−(λ−μ)^{2})a}{4γ} + \frac{γb}{2a} & 0 \end{matrix}) , λ=0,

where γ ≠ 0. Proposition 1.3^{1}.13. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.13 has the form:

λ = −\frac{1}{2}

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & γ \\ 0 & −α & 0 & \frac{2αd−γb}{2a} \\ −α & −γ & −\frac{2αd−γb}{2a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ = \frac{3}{2}

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{γb}{2a} \\ 0 & −γ & \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ ∉ \{−\frac{1}{2}, \frac{3}{2}\}

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & \frac{(1−(λ− \frac{1}{2} )^{2})a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & −\frac{(1−(λ− \frac{1}{2} )^{2})a}{4γ} + \frac{γb}{2a} & 0 \end{matrix}) , λ=0,

where γ ≠ 0. Proposition 1.3^{1}.14. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.14 has the form:

λ = 0

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & γ \\ 0 & −α & 0 & \frac{2αd−γb}{2a} \\ −α & −γ & −\frac{2αd−γb}{2a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ = 1

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{γb}{2a} \\ 0 & −γ & \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ ∉ {0, 1}

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & \frac{(1−(2λ−1)^{2})a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & −\frac{(1−(2λ−1)^{2})a}{4γ} + \frac{γb}{2a} & 0 \end{matrix}) , λ=0,

where γ ≠ 0.

Proposition 1.3^{1}.15. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.15.

Proposition 1.3^{1}.16. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.16.

Proposition 1.3^{1}.17. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.17 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & γ \\ 0 & −α & 0 & \frac{2αd−γb}{2a} \\ −α & −γ & −\frac{2αd−γb}{2a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

Proposition 1.3^{1}.18. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.18 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & γ \\ 0 & −α & 0 & \frac{2αd−γb}{2a} \\ −α & −γ & −\frac{2αd−γb}{2a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

Proposition 1.3^{1}.19. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.19 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & \frac{a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & −\frac{a}{4γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

Proposition 1.3^{1}.20. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.20 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & \frac{a}{4γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0.

Proposition 1.3^{1}.21. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.21 has the form:

λ = 0

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{γb}{2a} \\ 0 & −γ & \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ = 2

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & γ \\ 0 & −α & 0 & \frac{2αd−γb}{2a} \\ −α & −γ & −\frac{2αd−γb}{2a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

λ ∉ {0, 2}

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & \frac{(2λ−λ^{2})a}{4γ} − \frac{γb}{2a} \\ 0 & −γ & −\frac{(2λ−λ^{2})a}{4γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

Proposition 1.3^{1}.22. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.22 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & \frac{3a}{16γ} − \frac{γb}{2a} \\ 0 & −γ & −\frac{3a}{16γ} + \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0.

Proposition 1.3^{1}.23. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.23 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ \\ 0 & 0 & 0 & −\frac{γb}{2a} \\ 0 & −γ & \frac{γb}{2a} & 0 \end{matrix} ) , λ = 0,

where γ ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

Proposition 1.3^{1}.24. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.24 has the form:

λ = 0

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & 2γ & 0 \\ 0 & 0 & 0 & γ \\ −2γ & 0 & 0 & −\frac{γ(b+2c)}{2a} − \frac{a}{2γ} \\ 0 & −γ & \frac{γ(b+2c)}{2a} + \frac{a}{2γ} & 0 \end{matrix} ) , λ= \frac{2γ^{2}}{a^{2}} ,

where γ ≠ 0.

λ = 2

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & −2γ & α \\ 0 & 0 & α & γ \\ 2γ & −α & 0 & \frac{αd}{a} + \frac{γ(2c−b)}{2a} \\ −α & −γ & −\frac{αd}{a} − \frac{γ(2c−b)}{2a} & 0 \end{matrix}) , λ=−\frac{α^{2} + 2γ^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

For other values of λ the Einstein–Maxwell equations has no solutions.

Proposition 1.3^{1}.25. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.25 has the form:

λ = 0

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & −2γ & 0 \\ 0 & 0 & 0 & γ \\ 2γ & 0 & 0 & \frac{γ(2c−b)}{2a} + \frac{a}{2γ} \\ 0 & −γ & −\frac{γ(2c−b)}{2a} − \frac{a}{2γ} & 0 \end{matrix} ) , λ=−\frac{2γ^{2}}{a^{2}} ,

where γ ≠ 0.

λ = 2

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & 2γ & α \\ 0 & 0 & α & γ \\ −2γ & −α & 0 & \frac{αd}{a} − \frac{γ(2c+b)}{2a} \\ −α & −γ & −\frac{αd}{a} + \frac{γ(2c+b)}{2a} & 0 \end{matrix}) , λ= \frac{2γ^{2} − α^{2}}{a^{2}} ,

where α^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

For other values of λ the Einstein–Maxwell equations has no solutions.

Proposition 1.3^{1}.26. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.26.

Proposition 1.3^{1}.27. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.27.

Proposition 1.3^{1}.28. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.28.

Proposition 1.3^{1}.29. The Einstein–Maxwell equation has no solutions on the pair 1.3^{1}.29.

Proposition 1.3^{1}.30. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.30 has the form:

λ = 1, μ = 1

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & α \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & \frac{αd}{a} \\ −α & 0 & −\frac{αd}{a} & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

For other values of λ, μ the Einstein–Maxwell equation has no solutions.

Proposition 1.3^{1}.31. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.31 has the form:

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & β & α \\ 0 & 0 & α & γ \\ −β & −α & 0 & \frac{2αd−γb−βc}{2a} \\ −α & −γ & −\frac{2αd−γb−βc}{2a} & 0 \end{matrix}) , λ= \frac{βγ − α^{2}}{a^{2}} ,

where α^{2} + β^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

Proposition 1.3^{1}.32. Any solution of the Einstein–Maxwell equation on the pair 1.3^{1}.32 has the form:

B =( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω=(\begin{matrix}0 & 0 & β & α \\ 0 & 0 & α & γ \\ −β & −α & 0 & \frac{2αd−γb−βc}{2a} \\ −α & −γ & −\frac{2αd−γb−βc}{2a} & 0 \end{matrix}) , λ= \frac{βγ − α^{2}}{a^{2}} ,

where α^{2} + β^{2} + γ^{2} ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & d \\ a & 0 & d & c \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & −δ & 0 \end{matrix} ) , λ = 0.

Proposition 1.4^{1}.1. The Einstein–Maxwell equation has no solutions on the pair 1.4^{1}.1.

Proposition 1.4^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.2 has the form:

p = \frac{3}{2}

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = −\frac{3}{4d},

where \frac{b}{2d} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

p = \frac{5}{3}

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = 0, λ = −\frac{4}{3d}.

p ∉ \{\frac{3}{2}, \frac{5}{3}\}

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & 0 & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = 0, λ = −\frac{3(p − 1)^{2}}{d} .

Proposition 1.4^{1}.3. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.3 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & d & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = 0, λ = −\frac{3}{d}.

Proposition 1.4^{1}.4. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.4 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & −d & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = 0, λ = −\frac{3}{d}.

Proposition 1.4^{1}.5. The Einstein–Maxwell equation has no solutions on the pair 1.4^{1}.5.

Proposition 1.4^{1}.6. The Einstein–Maxwell equation has no solutions on the pair 1.4^{1}.6.

Proposition 1.4^{1}.7. The Einstein–Maxwell equation has no solutions on the pair 1.4^{1}.7.

Proposition 1.4^{1}.8. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.8 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = 0, λ = −\frac{3}{d}.

Proposition 1.4^{1}.9. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.9 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where p^{2} + p + r + \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.10. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.10 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where p^{2} + p + r = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.11. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.11 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where r + \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.12. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.12 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where r = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.13. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.13 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where r + 1 + \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.14. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.14 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where r + 1 = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.15. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.15 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 1 + \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.16. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.16 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{d}{2a} − 1 = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.17. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.17 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.18. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.18 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 1 + \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.19. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.19 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{d}{2a} − 1 = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.20. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.20 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{d}{2a} = \frac{α^{2}}{a} + \frac{β^{2}}{d}.

Proposition 1.4^{1}.21. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.21 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{α^{2}}{a} + \frac{β^{2}}{d} = 1.

Proposition 1.4^{1}.22. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.22 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{α^{2}}{a} + \frac{β^{2}}{d} = −1.

Proposition 1.4^{1}.23. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.23 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{α^{2}}{a} + \frac{β^{2}}{d} = 0.

Proposition 1.4^{1}.24. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.24 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{α^{2}}{a} + \frac{β^{2}}{d} = 1.

Proposition 1.4^{1}.25. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.25 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{α^{2}}{a} + \frac{β^{2}}{d} = −1.

Proposition 1.4^{1}.26. Any solution of the Einstein–Maxwell equation on the pair 1.4^{1}.26 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & b & c \\ 0 & 0 & c & d \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where \frac{α^{2}}{a} + \frac{β^{2}}{d} = 0.

Proposition 2.1^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 2.1^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & b \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{β^{2} + b}{b^{2}},

where α^{2} = \frac{b+β^{2}}{b^{2}} a^{2} − a.

Proposition 2.1^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 2.1^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & b \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}}{b^{2}} a^{2} − a.

Proposition 2.1^{1}.3. Any solution of the Einstein–Maxwell equation on the pair 2.1^{1}.3 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & b \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}}{b^{2}} a^{2}.

Proposition 2.1^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 2.1^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2} + a}{a^{2}},

where β^{2} = −\frac{α^{2}+a}{a^{2}}b^{2} − b.

Proposition 2.1^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 2.1^{2}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2} + a}{a^{2}},

where β^{2} = −\frac{α^{2}+a}{a^{2}}b^{2} + b.

Proposition 2.1^{2}.3. Any solution of the Einstein–Maxwell equation on the pair 2.1^{2}.3 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2} + a}{a^{2}},

where β^{2} = −\frac{α^{2}+a}{a^{2}}b^{2}.

Proposition 2.1^{2}.4. Any solution of the Einstein–Maxwell equation on the pair 2.1^{2}.4 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where β^{2} = −\frac{α^{2}}{a^{2}} b^{2} − b.

Proposition 2.1^{2}.5. Any solution of the Einstein–Maxwell equation on the pair 2.1^{2}.5 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where β^{2} = −\frac{α^{2}}{a^{2}} b^{2} + b.

Proposition 2.1^{2}.6. Any solution of the Einstein–Maxwell equation on the pair 2.1^{2}.6 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = 0.

Proposition 2.1^{3}.1. Any solution of the Einstein–Maxwell equation on the pair 2.1^{3}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} + b}{b^{2}},

where α^{2} = \frac{b+β^{2}}{b^{2}} a^{2} − a.

Proposition 2.1^{3}.2. Any solution of the Einstein–Maxwell equation on the pair 2.1^{3}.2 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{b^{2}},

where α^{2} = \frac{β^{2}−b}{b^{2}}a^{2} − a.

Proposition 2.1^{3}.3. Any solution of the Einstein–Maxwell equation on the pair 2.1^{3}.3 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2} − b}{b^{2}},

where α^{2} = \frac{β^{2}−b}{b^{2}}a^{2} + a.

Proposition 2.1^{3}.4. Any solution of the Einstein–Maxwell equation on the pair 2.1^{3}.4 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}}{b^{2}} a^{2} − a.

Proposition 2.1^{3}.5. Any solution of the Einstein–Maxwell equation on the pair 2.1^{3}.5 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}}{b^{2}} a^{2} + a.

Proposition 2.1^{3}.6. Any solution of the Einstein–Maxwell equation on the pair 2.1^{3}.6 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & β \\ −α & 0 & 0 & 0 \\ 0 & −β & 0 & 0 \end{matrix} ) , λ = \frac{β^{2}}{b^{2}} ,

where α^{2} = \frac{β^{2}}{b^{2}} a^{2}.

Proposition 2.1^{4}.1. Any solution of the Einstein–Maxwell equation on the pair 2.1^{4}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & b \\ 0 & 0 & b & −a \\ a & b & 0 & 0 \\ b & −a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & β \\ 0 & 0 & β & −α \\ −α & −β & 0 & 0 \\ −β & α & 0 & 0 \end{matrix} ) , λ = \frac{2a + α^{2} − β^{2}}{b^{2} − a^{2}} ,

where a^{2} ≠ b^{2}, αβ = \frac{β^{2}−α^{2}−2a}{b^{2}−a^{2}}ab − b, or

B = ( \begin{matrix}0 & 0 & a & εa \\ 0 & 0 & εa & −a \\ a & εa & 0 & 0 \\ εa & −a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & β \\ 0 & 0 & β & −α \\ −α & −β & 0 & 0 \\ −β & α & 0 & 0 \end{matrix} ) , λ = 2\frac{ε(α^{2} − β^{2}) − 2αβ}{ε(β^{2} − α^{2})^{2}} ,

where α^{2} ≠ β^{2}, a = \frac{β^{2}−α^{2}}{2} , ε ∈ \{−1, 1\}.

Proposition 2.1^{4}.2. Any solution of the Einstein–Maxwell equation on the pair 2.1^{4}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & b \\ 0 & 0 & b & −a \\ a & b & 0 & 0 \\ b & −a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & β \\ 0 & 0 & β & −α \\ −α & −β & 0 & 0 \\ −β & α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2} − β^{2}}{b^{2} − a^{2}} ,

where a^{2} ≠ b^{2}, αβ = \frac{β^{2}−α^{2}}{b^{2}−a^{2}} ab, or

B = ( \begin{matrix}0 & 0 & a & εa \\ 0 & 0 & εa & −a \\ a & εa & 0 & 0 \\ εa & −a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & δα \\ 0 & 0 & δα & −α \\ −α & −δα & 0 & 0 \\ −δα & α & 0 & 0 \end{matrix} ) , λ = −\frac{δα^{2}}{εa^{2}} ,

where ε ∈ {−1, 1}, δ ∈ {−1, 1}.

Proposition 2.2^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 2.2^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{3b − 2α^{2}}{2a^{2}} .

Proposition 2.2^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 2.2^{1}.2 has the form:

p = 2

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} .

p = −2

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

For other values of p the Einstein–Maxwell equation has no solutions.

Proposition 2.2^{1}.3. The Einstein–Maxwell equation has no solutions on the pair 2.2^{1}.3.

Proposition 2.2^{1}.4. Any solution of the Einstein–Maxwell equation on the pair 2.2^{1}.4 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & \frac{α^{2}b−a^{2}}{αa} & α \\ −α & −\frac{α^{2}b−a^{2}}{αa} & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α ≠ 0.

Proposition 2.2^{1}.5. Any solution of the Einstein–Maxwell equation on the pair 2.2^{1}.5 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & \frac{αb}{a} & α \\ −α & −\frac{αb}{a} & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & β & 0 \\ 0 & −β & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = 0.

Proposition 2.2^{1}.6. Any solution of the Einstein–Maxwell equation on the pair 2.2^{1}.6 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} .

Proposition 2.2^{1}.7. Any solution of the Einstein–Maxwell equation on the pair 2.2^{1}.7 has the form:

λ = 0

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & b & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} .

λ = 1

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & \frac{αb}{a} & α \\ −α & −\frac{αb}{a} & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & b & a \\ a & b & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & β & 0 \\ 0 & −β & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = 0.

λ = −1

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & β & α & 0 \\ −β & 0 & 0 & α \\ −α & 0 & 0 & γ \\ 0 & −α & −γ & 0 \end{matrix} ) , λ = \frac{βγ − α^{2}}{a^{2}} .

λ ∈] − 1, 1[\{0}

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} .

Proposition 2.2^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 2.2^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & 0 \\ a & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & −\frac{α^{2}b+a^{2}}{αa} \\ 0 & −α & \frac{α^{2}b+a^{2}}{αa} & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} ,

where α ≠ 0.

Proposition 2.2^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 2.2^{2}.2 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & 0 \\ a & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & \frac{a^{2}−α^{2}b}{αa} \\ 0 & −α & −\frac{a^{2}−α^{2}b}{αa} & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} ,

where α ≠ 0.

Proposition 2.2^{2}.3. Any solution of the Einstein–Maxwell equation on the pair 2.2^{2}.3 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & 0 \\ a & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & −\frac{αb}{a} \\ 0 & −α & \frac{αb}{a} & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & 0 \\ a & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0.

Proposition 2.2^{2}.4. Any solution of the Einstein–Maxwell equation on the pair 2.2^{2}.4 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & 0 \\ a & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & −\frac{αb}{a} \\ 0 & −α & \frac{αb}{a} & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} ,

where α ≠ 0, or

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & b & 0 \\ a & 0 & 0 & b \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0.

Proposition 2.2^{3}.1. Any solution of the Einstein–Maxwell equation on the pair 2.2^{3}.1 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & 0 & 0 \\ a & 0 & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} .

Proposition 2.3^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 2.3^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & α \\ 0 & 0 & −α & 0 \end{matrix} ) , λ = 0.

Proposition 2.4^{1}.1. The Einstein–Maxwell equation has no solutions on the pair 2.4^{1}.1.

Proposition 2.4^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 2.4^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = −\frac{3}{b}.

Proposition 2.4^{1}.3. Any solution of the Einstein–Maxwell equation on the pair 2.4^{1}.3 has the form:

B = ( \begin{matrix}0 & 0 & −a & 0 \\ 0 & a & 0 & 0 \\ −a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = 0.

Proposition 2.5^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = −\frac{6}{a}.

Proposition 2.5^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = 0.

Proposition 2.5^{1}.3. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.3 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where h − \frac{h^{2}}{2} + 2g = \frac{2αβ}{a}.

Proposition 2.5^{1}.4. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.4 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where h − \frac{h^{2}}{2} + 2g = \frac{2αβ}{a}.

Proposition 2.5^{1}.5. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.5 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 2g − \frac{1}{2} = \frac{2αβ}{a}.

Proposition 2.5^{1}.6. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.6 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 2g − \frac{1}{2} = \frac{2αβ}{a}.

Proposition 2.5^{1}.7. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.7 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & \frac{a}{α} \\ 0 & 0 & −\frac{a}{α} & 0 \end{matrix} ) , λ = 0,

where α ≠ 0.

Proposition 2.5^{1}.8. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.8 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & −\frac{a}{α} \\ 0 & 0 & \frac{a}{α} & 0 \end{matrix} ) , λ = 0,

where α ≠ 0.

Proposition 2.5^{1}.9. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.9 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & \frac{a}{α} \\ 0 & 0 & −\frac{a}{α} & 0 \end{matrix} ) , λ = 0,

where α ≠ 0.

Proposition 2.5^{1}.10. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.10 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & −\frac{a}{α} \\ 0 & 0 & \frac{a}{α} & 0 \end{matrix} ) , λ = 0,

where α ≠ 0.

Proposition 2.5^{1}.11. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.11 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where αβ = 0.

Proposition 2.5^{1}.12. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.12 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where αβ = 0.

Proposition 2.5^{1}.13. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.13 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where αβ = 0.

Proposition 2.5^{1}.14. Any solution of the Einstein–Maxwell equation on the pair 2.5^{1}.14 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where αβ = 0.

Proposition 2.5^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = −\frac{3}{a}.

Proposition 2.5^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 2r^{2} + 2p = −\frac{α^{2}+β^{2}}{a}.

Proposition 2.5^{2}.3. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.3 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 2r − \frac{1}{2} = \frac{α^{2}+β^{2}}{a}.

Proposition 2.5^{2}.4. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.4 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 2a = −(α^{2} + β^{2}), α^{2} + β^{2} ≠ 0.

Proposition 2.5^{2}.5. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.5 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & β \\ 0 & 0 & −β & 0 \end{matrix} ) , λ = 0,

where 2a = α^{2} + β^{2}, α^{2} + β^{2} ≠ 0.

Proposition 2.5^{2}.6. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.6 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

Proposition 2.5^{2}.7. Any solution of the Einstein–Maxwell equation on the pair 2.5^{2}.7 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.1^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 3.1^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} .

Proposition 3.1^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 3.1^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & −a & 0 \\ 0 & −a & 0 & 0 \\ a & 0 & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} .

Proposition 3.2^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 3.2^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = −\frac{6}{a}.

Proposition 3.2^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 3.2^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.2^{1}.3. Any solution of the Einstein–Maxwell equation on the pair 3.2^{1}.3 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = 0.

Proposition 3.2^{1}.4. Any solution of the Einstein–Maxwell equation on the pair 3.2^{1}.4 has the form:

λ = 1

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & −α & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , λ = 0.

λ ≠ 1

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.2^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 3.2^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = −\frac{3}{a}.

Proposition 3.2^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 3.2^{2}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.3^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 3.3^{1}.1 has the form:

p = 0

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

For other values of p the Einstein–Maxwell equation has no solutions.

Proposition 3.3^{1}.2. The Einstein–Maxwell equation has no solutions on the pair 3.3^{1}.2.

Proposition 3.3^{1}.3. The Einstein–Maxwell equation has no solutions on the pair 3.3^{1}.3.

Proposition 3.3^{1}.4. Any solution of the Einstein–Maxwell equation on the pair 3.3^{1}.4 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & b & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.3^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 3.3^{2}.1 has the form:

p = 0

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

For other values of p the Einstein–Maxwell equation has no solutions.

Proposition 3.3^{2}.2. The Einstein–Maxwell equation has no solutions on the pair 3.3^{2}.2.

Proposition 3.3^{2}.3. The Einstein–Maxwell equation has no solutions on the pair 3.3^{2}.3.

Proposition 3.3^{2}.4. Any solution of the Einstein–Maxwell equation on the pair 3.3^{2}.4 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & b & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.4^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 3.4^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & β & α & 0 \\ −β & 0 & 0 & α \\ −α & 0 & 0 & γ \\ 0 & −α & −γ & 0 \end{matrix} ) , λ = \frac{βγ − α^{2}}{a^{2}} .

Proposition 3.4^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 3.4^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & α & β & γ \\ −α & 0 & −γ & β \\ −β & γ & 0 & −α \\ −γ & −β & α & 0 \end{matrix} ) , λ = \frac{α^{2} + β^{2} + γ^{2}}{a^{2}} .

Proposition 3.5^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 3.5^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & 2a & 0 \\ 0 & a & 0 & 0 \\ 2a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = −\frac{3}{b}.

Proposition 3.5^{1}.2. The Einstein–Maxwell equation has no solutions on the pair 3.5^{1}.2.

Proposition 3.5^{1}.3. The Einstein–Maxwell equation has no solutions on the pair 3.5^{1}.3.

Proposition 3.5^{1}.4. Any solution of the Einstein–Maxwell equation on the pair 3.5^{1}.4 has the form:

B = ( \begin{matrix}0 & 0 & 2a & 0 \\ 0 & a & 0 & 0 \\ 2a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = 0.

Proposition 3.5^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 3.5^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = −\frac{3}{b}.

Proposition 3.5^{2}.2. The Einstein–Maxwell equation has no solutions on the pair 3.5^{2}.2.

Proposition 3.5^{2}.3. The Einstein–Maxwell equation has no solutions on the pair 3.5^{2}.3.

Proposition 3.5^{2}.4. Any solution of the Einstein–Maxwell equation on the pair 3.5^{2}.4 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix} ) , Ω = 0, λ = 0.

Proposition 4.1^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 4.1^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 4.1^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 4.1^{2}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

Proposition 4.2^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 4.2^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{6a + α^{2}}{a^{2}} .

Proposition 4.2^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 4.2^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = −\frac{α^{2}}{a^{2}} .

Proposition 4.2^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 4.2^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{6a + α^{2}}{a^{2}} .

Proposition 4.2^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 4.2^{2}.2 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2} − 6a}{a^{2}}.

Proposition 4.2^{2}.3. Any solution of the Einstein–Maxwell equation on the pair 4.2^{2}.3 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & α \\ −α & 0 & 0 & 0 \\ 0 & −α & 0 & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} .

Proposition 4.2^{3}.1. Any solution of the Einstein–Maxwell equation on the pair 4.2^{3}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & α & 0 & 0 \\ −α & 0 & 0 & 0 \\ 0 & 0 & 0 & α \\ 0 & 0 & −α & 0 \end{matrix} ) , λ = \frac{α^{2} − 6a}{a^{2}}.

Proposition 4.2^{3}.2. Any solution of the Einstein–Maxwell equation on the pair 4.2^{3}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & α & 0 & 0 \\ −α & 0 & 0 & 0 \\ 0 & 0 & 0 & α \\ 0 & 0 & −α & 0 \end{matrix} ) , λ = \frac{α^{2}}{a^{2}} .

Proposition 4.3^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 4.3^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & α \\ 0 & 0 & −α & 0 \end{matrix} ) , λ = 0.

Proposition 4.3^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 4.3^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & α \\ 0 & 0 & −α & 0 \end{matrix} ) , λ = 0.

Proposition 5.1^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 5.1^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 6.1^{1}.1. Any solution of the Einstein–Maxwell equation on the pair 6.1^{1}.1 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = −\frac{6}{a}.

Proposition 6.1^{1}.2. Any solution of the Einstein–Maxwell equation on the pair 6.1^{1}.2 has the form:

B = ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \end{matrix} ) , Ω = 0, λ = 0.

Proposition 6.1^{2}.1. Any solution of the Einstein–Maxwell equation on the pair 6.1^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = −\frac{3}{a}.

Proposition 6.1^{2}.2. Any solution of the Einstein–Maxwell equation on the pair 6.1^{2}.2 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = \frac{3}{a}.

Proposition 6.1^{2}.3. Any solution of the Einstein–Maxwell equation on the pair 6.1^{2}.3 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix} ) , Ω = 0, λ = 0.

Proposition 6.1^{3}.1. Any solution of the Einstein–Maxwell equation on the pair 6.1^{3}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & −a \end{matrix} ) , Ω = 0, λ = −\frac{3}{a}.

Proposition 6.1^{3}.2. Any solution of the Einstein–Maxwell equation on the pair 6.1^{3}.2 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & −a \end{matrix} ) , Ω = 0, λ = \frac{3}{a}.

Proposition 6.1^{3}.3. Any solution of the Einstein–Maxwell equation on the pair 6.1^{3}.3 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & −a \end{matrix} ) , Ω = 0, λ = 0.

2. Four-dimensional Einstein–Maxwell homogeneous spaces

A pseudo-Riemannian homogeneous space (\overline{G}, M) is said to be an Einstein–Maxwell homogeneous space if the Einstein–Maxwell equation has a solution on (\overline{G}, M). The pair (𝔤̄, 𝔤) corresponding to an Einstein–Maxwell homogeneous space (\overline{G}, M) will be called an Einstein–Maxwell pair. We can divide the four-dimensional Einstein–Maxwell homogeneous spaces (the Einstein–Maxwell pairs of codimension 4) into the Riemannian (with an invariant Riemannian metric), the Lorentzian (with an invariant pseudo-Riemannian metric of signature (3,1)), and the spaces (pairs) of type (2,2) (with an invariant pseudo-Riemannian metric of signature (2,2)). Using the results of Chapter II, 1, we can prove the following Theorems:

Theorem 1. Any Riemannian Einstein–Maxwell pair (𝔤̄, 𝔤) of codimension 4 is equivalent to one and only one of the following pairs: 1.1^{2}.1, 1.1^{2}.2 (p = 1), 1.1^{2}.6–1.1^{2}.12, 2.1^{3}.1–2.1^{3}.6, 3.4^{2}.1, 3.5^{2}.1, 3.5^{2}.4, 4.2^{2}.1–4.2^{2}.3, 6.1^{2}.1–6.1^{2}.3.

Theorem 2. Any Lorentzian Einstein–Maxwell pair (𝔤̄, 𝔤) of codimension 4 is equivalent to one and only one of the following pairs: 1.1^{1}.2 (p = \frac{1}{2}), 1.1^{1}.5–1.1^{1}.7, 1.1^{1}.10 (λ = 0), 1.1^{2}.2 (p = 1), 1.1^{2}.6–1.1^{2}.10, 1.1^{2}.12 (λ = 0), 1.1^{3}.1, 1.1^{4}.1, 1.4^{1}.2–1.4^{1}.4, 1.4^{1}.8– 1.4^{1}.26, 2.1^{2}.1–2.1^{2}.6, 2.4^{1}.2, 2.4^{1}.3, 2.5^{2}.1–2.5^{2}.7, 3.2^{2}.1, 3.2^{2}.2, 3.3^{2}.1 (p = 0), 3.3^{2}.4, 3.5^{1}.1, 3.5^{1}.4, 3.5^{2}.1, 3.5^{2}.4, 4.1^{2}.1, 6.1^{3}.1–6.1^{3}.3.

Theorem 3. Any Einstein–Maxwell pair (𝔤̄, 𝔤) of codimension 4 with an invariant pseudo-Riemannian metric of signature (2,2) is equivalent to one and only one of the following pairs: 1.1^{1}.1, 1.1^{1}.2 (p = \frac{1}{2}), 1.1^{1}.5–1.1^{1}.10, 1.1^{2}.1, 1.1^{2}.2 (p = 1), 1.1^{2}.6– 1.1^{2}.12, 1.1^{5}.1, 1.1^{6}.1, 1.2^{1}.1, 1.2^{2}.1, 1.3^{1}.1, 1.3^{1}.5 (λ = 0, μ ∈ {−2, 2}), 1.3^{1}.8– 1.3^{1}.14, 1.3^{1}.17–1.3^{1}.23, 1.3^{1}.24 (λ ∈ {0, 2}), 1.3^{1}.25 (λ ∈ {0, 2}), 1.3^{1}.30 (λ = μ = 1), 1.3^{1}.31, 1.3^{1}.32, 1.4^{1}.2–1.4^{1}.4, 1.4^{1}.8–1.4^{1}.26, 2.1^{1}.1–2.1^{1}.3, 2.1^{3}.1–2.1^{3}.6, 2.1^{4}.1, 2.1^{4}.2, 2.2^{1}.1, 2.2^{1}.2 (p ∈ {−2, 2}), 2.2^{1}.4–2.2^{1}.7, 2.2^{2}.1–2.2^{2}.4, 2.2^{3}.1, 2.3^{1}.1, 2.4^{1}.2, 2.4^{1}.3, 2.5^{1}.1–2.5^{1}.14, 3.1^{1}.1, 3.1^{2}.1, 3.2^{1}.1–3.2^{1}.4, 3.3^{1}.1 (p = 0), 3.3^{1}.4, 3.4^{1}.1, 3.5^{1}.1, 3.5^{1}.4, 4.1^{1}.1, 4.2^{1}.1, 4.2^{1}.2, 4.2^{3}.1, 4.2^{3}.2, 4.3^{1}.1, 4.3^{1}.2, 5.1^{1}.1, 6.1^{1}.1, 6.1^{1}.2.

The solution of the Einstein–Maxwell equation on the pair m.n^{k}.i of a fixed signature can be found from Proposition m.n^{k}.i. In addition, the metric B satisfies the following complementary conditions:

1. If the signature of B is (3,1) (or (1,3)), then det B < 0.

2. If the signature of B is (4,0) (or (0,4)), then det B > 0 and B is positive (or negative) definite.

3. If the signature of B is (2,2), then det B > 0 and B is neither positive, nor negative definite.

Example. Consider the pair 1.1^{2}.1. The invariant pseudo-Riemannian metric on the pair 1.1^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) , det B = a^{2}(bd − c^{2}).

It follows that the pair 1.1^{2}.1 admits an invariant pseudo-Riemannian metric of arbitrary signature, i.e., the pair 1.1^{2}.1 is a Riemannian pair if bd − c^{2} > 0, ab > 0, a Lorentzian pair if bd − c^{2} < 0, and a pair of type (2,2) if bd − c^{2} > 0, ab < 0.

By Proposition 1.1^{2}.1, any solution of the Einstein–Maxwell equation on the pair 1.1^{2}.1 has the form:

B = ( \begin{matrix}a & 0 & 0 & 0 \\ 0 & b & 0 & c \\ 0 & 0 & a & 0 \\ 0 & c & 0 & d \end{matrix} ) ,

where 4a^{2} = bd − c^{2},

Ω = ( \begin{matrix}0 & 0 & α & 0 \\ 0 & 0 & 0 & −2α \\ −α & 0 & 0 & 0 \\ 0 & 2α & 0 & 0 \end{matrix} ) , λ = \frac{2α^{2} − 3b}{2a^{2}}.

It follows that the Einstein–Maxwell equation on the pair 1.1^{2}.1 has a solution in the case of signatures (4,0) and (2,2), and has no solutions in the case of signature (3,1), i.e., the pair 1.1^{2}.1 is a Riemannian Einstein–Maxwell pair, as well as an Einstein–Maxwell pair of type (2,2).

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Belarussian State University, Department of Applied Mathematics and Informatics E-mail address: bk@islc.org

Received June, 15, 2001