On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices

Parametrization of $4\times 4$-matrices $G$ of the complex linear group $GL(4,C)$ in terms of four complex 4-vector parameters $(k,m,n,l)$ is investigated. Additional restrictions separating some subgroups of $GL(4,C)$ are given explicitly. In the given parametrization, the problem of inverting any $4\times 4$ matrix $G$ is solved. Expression for determinant of any matrix $G$ is found: $\det G = F(k,m,n,l)$. Unitarity conditions $G^{+} = G^{-1}$ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups $G_{1}$, $G_{2}$, $G_{3}$ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators $\Lambda_{k}$, being of Gell-Mann type, substantially differs from the basis $\lambda_{i}$ used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of $GL(4,C)$ can be used $\{\Lambda_k\} = \{\alpha_i\oplus\beta_j\oplus(\alpha_iV\beta_j = {\boldsymbol K} \oplus {\boldsymbol L}\oplus{\boldsymbol M})\}$, which permit to factorize SU(4) transformations according to $S = e^{i\vec{a}\vec{\alpha}} e^{i\vec{b}\vec\beta}} e^{i{\boldsymbol k}{\boldsymbol K}} e^{i{\boldsymbol l}{\boldsymbol L}} e^{i\boldsymbol m}{\boldsymbol M}}$, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups.

Because of so many applications in physics, various parametrizations for the group elements of unitary group SU (4) and related to it deserve special attention. Our efforts will be given to extending some classical technical approaches proving their effectiveness in simple cases of the linear and unitary groups SL(2, C) and SU (2), so that we will work with objects known by every physicist, such as Pauli and Dirac matrices. This paper, written for physicists, is self-contained in that it does not require any previous knowledge of the subject nor any advanced mathematics.
Let us start with the known example of spinor covering for complex Lorentz group: consider the 8-parametric 4 × 4 matrices in the quasi diagonal form [18,32,45] The composition rules for parameters k = (k 0 , k) and m = (m 0 , m) are With two additional constraints on 8 quantities k 2 0 − k 2 = +1, m 2 0 − m 2 = +1, we will arrive at a definite way to parameterize a double (spinor) covering for complex Lorentz group SO (4, C). At this, the problem of inverting of the G matrices with unit determinant det G is solved straightforwardly: G = G(k 0 , k, m 0 , m), G −1 = G(k 0 , −k, m 0 , −m). Transition from covering 4-spinor transformations to 4-vector ones is performed through the known relationship Gγ a G −1 = γ c L a c which determine 2 =⇒ 1 map from ±G to L.
There exists a direct connection between the above 4-dimensional vector parametrization of the spinor group G(k a , m a ) and the Fedorov's parametrization [32] of the group of complex orthogonal Lorentz transformations in terms of 3-dimensional vectors Q = k/k 0 , M = m/m 0 , with the simple composition rules for vector parameters Evidently, the pair (Q, M ) provides us with possibility to parameterize correctly orthogonal matrices only. Instead, the (k a , m a ) represent correct parameters for the spinor covering group. When we are interested only in local properties of the spinor representations, no substantial differences between orthogonal groups and their spinor coverings exist. However, in opposite cases global difference between orthogonal and spinor groups may be very substantial as well as correct parametrization of them.
To parameterize 4-spinor and 4-vector transformations of the complex Lorentz group one may use curvilinear coordinates. The simplest and widely used ones are Euler's complex angles (see [32] and references in [18]). In general, on the basis of the analysis given by Olevskiy [58] about coordinates in the real Lobachevski space, one can propose 34 different complex coordinate systems appropriate to parameterize the complex Lorentz group and its double covering.
A particular, Euler angle parametrization is closely connected with cylindrical coordinates on the complex 3-sphere, one of 34 possible coordinates. Such complex cylindrical coordinates can be introduced by the following relations [18]: k 0 = cos ρ cos z, k 3 = i cos ρ sin z, k 1 = i sin ρ cos φ, k 2 = i sin ρ sin φ, m 0 = cos R cos Z, m 3 = i cos R sin Z, m 1 = i sin RΦ, m 2 = i sin R sin Φ.
The main question is how to extend possible parameterizations of small orthogonal group SO(4, C) and its double covering to bigger orthogonal and unitary groups 1 . To be concrete we are going to focus attention mainly on the group SU (4) and its counterparts SU (2, 2), SU (3,1).
There exist many publications on the subject, a great deal of facts are known -in the following we will be turning to them. A good classification of different approaches in parameterizing finite transformations of SU (4) was done in the recent paper by A. Gsponer [35]. Recalling it, we will try to cite publications in appropriate places though many of them should be placed in several different subclasses -it is natural because all approaches are closely connected to each other.
with the hope that exp n (i) could be summed in closed form and also that these factors have simple properties. This possibility for the groups SU (4) and SU (2, 2) will be discussed in more detail in sections below.
• Product form [12,13,14,16,27,38,56,57]. An extreme non-canonical form is to factorize the general exponential into a product of n simplest 1-parametric exponentials • Basic elements (the main approach in the present treatment) [8,9,35,40,44,45,46,47,74]. This way is to expand the elements of the group (matrices or quaternions) into a sum over basis elements and to work with a linear decomposition of the matrices over basic ones: as by definition the relationships λ m λ n = e mnk λ k must hold, the group multiplication rule for parameters x k looks The main claim is that the all properties of any matrix group are straightforwardly determined by the bilinear function, the latter is described by structure constants e mnk entering the multiplication rule λ m λ n = e mnk λ k .
In our opinion, we should search the most simplicity in mathematical sense while working with basic elements λ k and the structure constants determining the group multiplication rule (1.1), (1.2).
The material of this paper is arranged as follows.
In Section 2 an arbitrary 4 × 4 matrix G ∈ GL(4, C) is decomposed into sixteen Dirac matrices 3 for definiteness we will use the Weyl spinor basis; four 4-dimensional vectors (k, m, l, n) are definite linear combinations of A, B, A l , B l , F mn -see (2.4). In such parameters (2.3), the group multiplication law G ′′ = G ′ G is found in explicit form.
Then we turn to the following problem: at given G = G(k, m, n, l) one should find parameters of the inverse matrix: G −1 = G(k ′ , m ′ , n ′ , l ′ ) -expressions for (k ′ , m ′ , n ′ , l ′ ) have been found explicitly (for details of calculation see [62]). Also, several equivalent expressions for determinant det G have been obtained, which is essential when going to special groups SL(4, C) and its subgroups.
In Section 3, with the help of the expression for the inverse matrix G −1 (k ′ , m ′ , l ′ , n ′ ) we begin to consider the unitary group SU (4). To this end, one should specify the requirement of unitarity G + = G −1 to the above vector parametrization -so that unitarity conditions are given as non-linear cubic algebraic equations for parameters (k, m, l, n) including complex conjugation.
In Section 4 we have constructed three 2-parametric solutions of the produced equations of unitarity 4 , these subgroups G 1 , G 2 , G 3 consist of two commuting Abelian unitary subgroups.
The task of complete solving of the unitarity conditions seems to be rather complicated. In remaining part of the present paper we describe some relations of the above treatment to other considerations of the problem in the literature. We hope that the full general solution of the unitary equations obtained can be constructed on the way of combining different techniques used in the theory of the unitary group SU (4) and it will be considered elsewhere.
We turn again to the explicit form of the Dirac basis and note that all 15 matrices are of Gell-Mann type: they have a zero-trace, they are Hermitian, besides their squares are unite: Exponential function of any of them equals to Evidently, multiplying such 15 elementary unitary matrices (at real parameters x i ) gives again an unitary matrix At this there arises one special possibility to determine extended Euler angles a 1 , . . . , a 15 . For the group SU (4) the Euler parametrization of that type was found in [69]. A method to solve the problem in [69] was based on the use yet known Euler parametrization for SU (3) -the latter problem was solved in [25]. Extension to SU (N ) group was done in [70,71]. Evident advantage of the Euler angles approach is its simplicity, and evident defect consists in the following: we do not know any simple group multiplication rule for these angles -even the known solution for SU (2) is too complicated and cannot be used effectively in calculation. In Section 6 the main question is how in Dirac parametrization one can distinguish SU (3), the subgroup in SU (4). In this connection, it should be noted that the basis λ i used in [25] substantially differs from the above Dirac basis Λ i -this peculiarity is closely connected with distinguishing the SU (3) in SU (4). In order to have possibility to compare two approaches we need exact connection between λ i and Λ i -we have found required formulas 6 . The separation of SL(3, C) in SL(4, C) is given explicitly, at this 3 × 3 matrix group is described with the help of 4 × 4 matrices 7 . The group law for parameters of SL(3, C) is specified.
In Section 7 one different way to list 15 generators of GL(4, C) is examined 8 these two set commute with each others α j β k = β k α j , and their multiplications provides us with 9 remaining basis elements of fifteen: We turn to the rule of multiplying 15 generators α i , β i , A i , B i , C i and derive its explicit form (see (7.3)). Section 8 adds some facts to a factorized structure of SU (4). To this end, between 9 generators we distinguish three sets of commuting ones an arbitrary element from GL(4, C) can be factorized as follows 9 where K, L, M are 3-parametric groups, each of them consists of three Abelian commuting unitary subgroups 10 . On the basis of 15 matrices one can easily see 20 ways to separate SU (2) subgroups, which might be used as bigger elementary blocks in constructing a general transformation 11 . In Sections 9 and 10 we specify our approach for pseudounitary groups SU (2, 2) and SU (3, 1) respectively. All generators Λ ′ k of these groups can readily be constructed on the basis of the known Dirac generators of SU (4) (see (9.1)).
2 On parameters of inverse transformations G −1 Arbitrary 4 × 4 matrix G ∈ GL(4, C) can be decomposed in terms of 16 Dirac matrices (such an approach to the group L(4, C) was discussed and partly developed in [5,8,9,6,45,48,49,50,66] and especially in [40]): Taking 16 coefficients A, B, A l , B l , F mn as parameters in the group G = G(A, B, A l , B l , F mn ) one can establish the corresponding multiplication law for these parameters: The latter formulas are correct in any basis for Dirac matrices. Below we will use mainly Weyl spinor basis: With this choice, let us make 3 + 1-splitting in all the formulas: where complex 4-vector parameters (k, l, m, n) are defined by [18]: For such parameters (2.3), the composition rule (2.2) will look as follows: Now let us turn to the following problem: with given G = G(k, m, n, l) one should find parameters of the inverse matrix: G −1 = G(k ′ , m ′ , n ′ , l ′ ). In other words, starting from G(k, m, n, l) = , one should calculate parameters of the inverse matrix G −1 . The problem turns to be rather complicated 12 , the final result is (D = det G, (mn) ≡ m 0 n 0 − mn, and so on) Substituting equations (2.7) into equation G −1 G = I one arrives at After calculation, one can prove these identities and find the determinant: D = det G(k, m, n, l) = (kk)(mm) + (ll)(nn) + 2(mk)(ln) + 2(lk)(nm) − 2(nk)(lm) Let us specify several more simple subgroups.

Case A
Let 0-components k 0 , m 0 , l 0 , n 0 be real-valued, and 3-vectors k, m, l, n be imaginary. Performing in (2.5) the formal change (new vectors are real-valued) then the multiplication rules (2.5) for sixteen real variables look as follows where the notation is used: [ab] = a 0 b 0 + ab.

Case C
In (2.9) one can impose additional restrictions m 0 = k 0 , l 0 = n 0 , m = −k, l = −n; (2.10) at this G(k 0 , k, n 0 , n) looks and the composition rule is

Case D
There exists one other subgroup defined by the composition law (2.5) becomes simpler as well as the determinant D det G = (kk)(mm).
If one additionally imposes two requirements (kk) = +1, (mm) = +1, the Case D describes spinor covering for special complex rotation group SO(4, C); this most simple case was considered in detail in [18]. It should be noted that the above general expression (2.8) for determinant can be transformed to a shorter form which for the three Cases A, B, C becomes yet simpler:

Unitarity condition
Now let us turn to consideration of the unitary group SU (4). One should specify the requirement of unitarity G + = G −1 to the above vector parametrization. Taking into account the formulas which can be represented differently we arrive at With the use of expressions for parameters of the inverse matrix with additional restriction det G = +1 equations (3.2) can be rewritten as Thus, the known form for parameters of the inverse matrix G −1 makes possible to write easily relations (3.3) representing the unitarity condition for group SU (4). Here there are 16 equations for 16 variables; evidently, not all of them are independent. Let us write down several simpler cases.

Case A
With formal change 13 Here there are 16 equations for 16 real-valued variables.

Case B
Let or symbolically m = k * , l = n * . The unitarity relations become and 8 conjugated ones It may be noted that latter relations are greatly simplified when n = 0, or when k = 0. Firstly, let us consider the case n = 0: Taking in mind the identity we arrive at k * 0 = +k 0 , k * = −k. It has sense to introduce the real-valued vector c a : Another possibility is realized when k = 0: With the use of identity we get Corresponding matrices G(0, 0, n, l = n * ) make up a special set of unitary matrices However, it must be noted that these matrices (3.5) do not provide us with any subgroup because

2-parametric subgroups in SU (4)
To be certain in correctness of the produced equations of unitarity, one should try to solve them at least in several most simple particular cases. For instance, let us turn to the Case C and specify equations (3.6) for a subgroup arising when k = (k 0 , k 1 , 0, 0) and n = (n 0 , n 1 , 0, 0): they are four non-linear equations for four real variables. It may be noted that equations (4.1) can be regarded as two eigenvalue problems in two dimensional space (with eigenvalue +1): The determinants in both problems must be equated to zero: The latter equations may be rewritten in factorized form: They have the structure: AC = 0, BC = 0. Four different cases arise.
(1) Let C = 0, then (2) Now, let A = 0, B = 0, but a contradiction arises: (3)-(4) There are two simples cases: Evidently, (4 which can be transformed to Both equations (4.2) and (4.5) are to be satisfied from where it follows They specify a 2-parametric unitary subgroup in SU (4) Two analogous subgroups are possible: . (4.8) Let us consider the latter subgroup (4.8) in some detail. The multiplication law for parameters is For two particular cases (see (4.3) and (4.4)), these formulas take the form: Therefore, multiplying of any two elements from G ′ 0 3 does not lead us to any element from G ′ 0 3 , instead belonging to G 0 : G 0 ′ 3 G 0 3 ∈ G 0 . Similar result would be achieved for G 1 and G 2 : In the subgroup given by (4.9) one can easily see the structure of the 1-parametric Abelian subgroup: In the same manner, similar curvilinear parametrization can be readily produced for 2parametric groups (4.6)-(4.8). For definiteness, for the subgroup G 3 such coordinates are given by k 0 = cos α cos ρ, k 3 = cos α sin ρ, (4.11) One may note that equation (4.11) at ρ = 0 will coincide with G 0 (α) in (4.10): G 3 (ρ = 0, α) = G 0 (α). Similar curvilinear parametrization may be introduced for two other subgroups, G 1 and G 2 .
One could try to obtain more general result just changing real valued curvilinear coordinates on complex. However it is easily verified that it is not the case: through that change though there arise subgroups but they are not unitary. Indeed, let the matrix (4.10) be complex: then unitarity condition gives cos α cos α * + sin α sin α * = 1, − cos α sin α * + sin α cos α * = 0.
These two equations can be satisfied only by a real valued α. In the same manner, the the formal change } again provides us with non-unitary subgroups. It should be noted that each of three 2-parametric subgroup G 1 , G 2 , G 3 , in addition to G 0 (α), contains one additional Abelian unitary subgroup: It may be easily verified that and with notation we arrive at

4-parametric unitary subgroup
Let us turn again to the subgroup in GL(4, C) given by Case C (see (2.10)): when the unitarity equations look as follows: They can be rewritten as four eigenvalue problems: These equations have the same structure which gives two different eigenvalues In explicit form, equations (5.1) looks as follows: The eigenvalue λ = +1 might be constructed by two ways: These two relations (5.4) are equivalent to the following one: Thus, equations (5.3) have two different types: Now let us turn to equations (5.2). They have the form As we are interested only in positive eigenvalue λ = +1, we must use only one possibility λ = +1 = λ 1 , so that Vector condition in (5.7) says that k and n are (anti)collinear: With notation (5.8), equations (5.5)-(5.6) take the form: where Therefore, we have 8 variables e, k 0 , n 0 , K, N and the set of equations, (5.9)-(5.11) for them. Its solving turns to be rather involving, so let us formulate only the final result: .
It should be noted that The unitarity of the matrices (5.12) may be verified by direct calculation. Indeed, and further for GG + = I we get (by 2 × 2 blocks) One different way to parameterize (5.12) can be proposed. Indeed, relations (5.12) are Therefore, matrix G can be presented as follows: Evidently, it suffices to take positive values for W . The constructed subgroup (5.13) depends upon four parameters k 0 , n 0 , e: Let us establish the law of multiplication for four parameters k 0 , n 0 , W = W e: or by 2 × 2 blocks As (11) = (22), (12) = −(21); further one can consider only two blocks: So the composition rules should be The later formula coincides with the Gibbs multiplication rule (see in [32]) for 3-dimensional rotation group SO(3, R). It remains to prove the identity: First terms are Second term is Therefore, (5.14) takes the form which is identity due to equalities It is matter of simple calculation to introduce curvilinear parameters for such an unitary subgroup: e = (sin θ cos φ, sin θ sin φ, cos θ), k 0 = cos α cos ρ, K = cos α sin ρ, n 0 = sin α cos ρ, N = − sin α sin ρ, and G looks as follows G = ∆ Σ −Σ ∆ , ∆ = cos α(cos ρ + i sin ρ cos θ) i cos α sin ρ sin θe −iφ i cos α sin ρ sin θe iφ cos α(cos ρ − i sin ρ cos θ , Σ = sin α(cos ρ + i sin ρ cos θ) +i sin α sin ρ sin θe −iφ i sin α sin ρ sin θe iφ sin α(cos ρ − i sin ρ cos θ) .
It should be noted that one one may factorize 4-parametric element into two unitary factors, 1-parametric and 3-parametric. Indeed, let us consider the product of commuting unitary groups, isomorphic to Abelian group G 0 and SU (2): takes the form The task of full solving of the unitarity conditions seems to be rather complicated and it will be considered elsewhere. In the remaining part of the present paper we describe some relations of the above treatment to other considerations of the problem in the literature. The relations described give grounds to hope that the full general solution of the unitary equations obtained can be constructed on the way of combining different techniques used in the theory of the unitary group SU (4). Exponential function of any of them equals to U = e iaΛ = cos a + i sin aΛ, det e iaΛ = +1, Evidently, multiplying of such 15 elementary unitary matrices (at real parameters x i ) results in an unitary matrix At this there arise 15 generalized angle-variables a 1 , . . . , a 15 . Evident advantage of this approach is its simplicity, and evident defect consists in the following: we do not know any simple group multiplication rule for these angles.
It should be noted that the basis λ i used in [69] substantially differs from the above Dirac basis Λ i -this peculiarity is closely connected with distinguishing SU (3) in SU (4). This problem is evidently related to the task of distinguishing GL(3, C) in GL(4, C) as well.

On the multiplication law for GL(4, C) in Dirac basis
In the Gell-Mann basis λ i , an element of GL(4, C) is or in variables (k, m, l, n): The problem is to establish the multiplication rule G ′′ = G ′ G in λ-basis: As by definition the relationships λ m λ n = e mnk λ k must hold, the multiplication rule is The main claim is that the all properties of the GL(4, C) with all its subgroups are determined by the bilinear function (7.1), the latter is described by structure constants e mnk . It is evident that these group constants should be simpler in the Dirac basis Λ i than in the basis λ i . Our next task is to establish the multiplication law G ′′ = G ′ G in Λ-basis: Before searching for structural constants E mnk , let us introduce a special way to list the Dirac basis Λ i : these two set commute with each others α j β k = β k α j , and their multiplications provides us with 9 remaining basis elements of {Λ k }: The multiplication rules for basic elements These relations provide us with simple formulas for fifteen coordinates of the element of GL(4, C) With the use of relations (7.3) an explicit form of the group law for (15 + 1) parameters can be found: From these relations we arrive at the following composition rules: With the help of the index notation it is easy to see a cyclic symmetry in the above relationships: It is readily seen that these group multiplication laws (7.4), (7.5) permit 15 two-parametric subgroups: with the same composition law: which in variables γ = W cos φ, a = iW sin φ takes the form The variable W is determined by det G(W, α) = W 4 , the choice W = 1 guarantees det G = +1. All 15 basis elements Λ (ρ) ∈ {α k , β k , A k , B k , C k } possess the same properties: Therefore, one can construct 15 different elementary unitary (at real valued parameters) matrices by one the same recipe: The whole set of unitary matrices SU (4) may be constructed on the basis of a simple factorized formula: (15) Λ (15) .
The order of the factors is important. Every such order leads us to a definite parametrization for the group SU (4) -all them seem to be equivalent.
In the end of the section let us write down the explicit form of these 15 elementary unitary transformations: Certainly, these relations provide us with 15 elementary solutions of the unitarity equations (3.3). For instance, the generator α 2 gives rise to the above 1-parametric Abelian subgroup G 0 (α); whereas the above 4-parametric subgroup G 0 × SU (2) (5.15) is generated by (α 2 ; β 1 , B 2 , C 2 ). The question is how one could describe all combinations of the above 15 simple sub-solutions by a single unifying formula -the latter should evidently exist.

On factorization SU (4) and the group fine-structure
On the basis of 9 matrices (7.2) one can construct six 3-dimensional sub-sets: (one may recall the rule to calculate the determinant of a 3 × 3 matrix) with the same commutation relations: When all parameters are real-valued, the formula provides us with the rule to construct elements from SU (4) group 14 . The order of factors might be different. Let us specify the group law for these 5 subsets. First are the two groups: e i a α = cos a + i sin a(n 1 α 1 + n 2 α 2 + n 3 α 3 ), They are isomorphic, so one can consider only the first one: Multiplying two matrices we arrive at Parameters (x 0 , x i ) should obey The inverse matrix looks With real (x 0 , x i ) we have a group isomorphic to SU (2), spinor covering for SO(3, R): At complex (x 0 , x i ) we have a group isomorphic to GL(2, C), spinor covering for SO(3, C) or Lorentz group. Now let us turn to finite transformations from remaining subsets. It is readily verified that these 1-parametric finite elements e iy 1 Γ 1 = cos y 1 + i sin y 1 Γ 1 , e iy 2 Γ 1 = cos y 2 + i sin y 2 Γ 2 , e iy 3 Γ 3 = cos y 3 + i sin y 3 Γ 3 , commute with each other: e iy 1 Γ 1 e iy 2 Γ 2 = (cos y 1 + i sin y 1 Γ 1 )(cos y 2 + i sin y 2 Γ 2 ) = = cos y 1 cos y 2 + i cos y 1 sin y 2 Γ 2 + i cos y 2 sin y 1 Γ 1 + sin y 1 sin y 2 Γ 3 , e iy 2 Γ 2 e iy 1 Γ 1 = (cos y 2 + i sin y 2 Γ 2 )(cos y 1 + i sin y 1 Γ 1 ) = = cos y 2 cos y 1 + i cos y 2 sin y 1 Γ 1 + i cos y 1 sin y 2 Γ 2 + sin y 2 sin y 1 Γ 3 , that is e iy 1 Γ 1 e iy 2 Γ 2 = e iy 2 Γ 2 e iy 1 Γ 1 , and so on. Evidently, this property correlates with the commutative relations (8.1). Thus, each of tree subgroups can be constructed as multiplying of elementary 1-parametric commuting transformations. Their explicit forms are: subgroup K e il 1 L 1 = cos l 1 + i sin l 1 C 1 , e il 2 L 1 = cos l 2 + i sin l 2 A 2 , e il 3 L 3 = cos l 3 + i sin l 3 B 3 ; subgroup M One additional note should be made. In the recent paper by A. Gsponer [35] on the quaternion approach to the problem of building the finite transformations from SU (3) and SU (4) an important point was to divide 15 basis 4 × 4 matrices into three sets: set A of antisymmetrical matrices, set S of symmetrical matrices, set D of diagonal traceless ones.
It is easily seen that which exactly corresponds to the structure used in [35] in connection with the Lanczos decomposition theorem [43].
Now let us turn to finite transformations from remaining sub-sets K ′ , L ′ , M ′ . Each of tree subgroups can be constructed as multiplying of elementary 1-parametric commuting transformations. Their explicit forms are: On the basis of 15 matrices

Discussion
Let us summarize the main point of the present treatment. Parametrization of 4 × 4 matrices G of the complex linear group GL(4, C) in terms of four complex 4-vector parameters (k, m, n, l) is investigated. Additional restrictions separating some subgroups of GL(4, C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F (k, m, n, l). Unitarity conditions G + = G −1 have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G 1 , G 2 , G 3 -each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU (2) and 1-parametric Abelian group.
The Dirac basis of generators Λ k , being of Gell-Mann type, substantially differs from the basis λ i used in the literature on SU (4) group, formulas relating them are found -they permit to separate SU (3) subgroup in SU (4). Special way to list 15 Dirac generators of GL(4, C) can be used {Λ k } = {α i ⊕ β j ⊕ (α i β j = K ⊕ L ⊕ M )}, which permit to factorize SU (4) transformations according to S = e i a α e i b β e ikK e ilL e imM , where two first factors commute with each other and are isomorphic to SU (2) group, the three last are 3-parametric groups, each of them consists of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λ k permits to separate twenty 3-parametric subgroups in SU (4) isomorphic to SU (2); those subgroups might be used as bigger elementary blocks in constructing a general transformation SU (4). It is shown how one can specify the present approach for the unitary group SU (2, 2) and SU (3, 1).
In principle, all different approaches used in the literature are closely related so that any result obtained within one technique may be easily translated to any other. There is no sense to persist in exploiting only one representation, thinking that it is much better than all others. Success should lie in combining different techniques. For instance, Euler angles-based approach provides us with the group elements in the separated variables form, which may be of a supreme importance at calculating matrix elements of the group. In turn, a factorized subgroup-based structure is of special interest in the particle physics and gauge theory of fundamental interaction. Geometrical properties of the groups, their global structure, differences between orthogonal groups and their double covering, and so on, seem to be most easily understood in terms of bilinear functions in space of linear parameters: G = x j Λ j , x ′′ j = e jkl x ′ k x l . We have no ground to think that only exponential functions e iΛ are suitable for exploration into group structures. We may expect that in addition to Euler angles many other curvilinear coordinates might be of value for studying of the group structure. For instance, in the case of the group SO(4, C) we have known 34 such coordinate systems owing to Olevskiy investigation [58] on 3-orthogonal coordinates in real Lobachevski space.
In conclusion, several words about possible application areas of the obtained results. The main argument in favor of constructing the theory of unitary groups SU (4) (and related to it) in terms of Dirac matrices is the role of spinor methods being widely adopted in physics. Let us mention several problems most attractive for authors: SU (2, 2) and conformal symmetry, massless particles; classical Yang-Mills equations and gauge fields; geometric phases for multi-level quantum systems; composite structure of quarks and leptons; SU (4) gauge models.
In particular, description of the group SU (2, 2) in terms of matrices α j , β j should be of great benefit in investigation of conformal symmetry in massless particles theory. For instance, classical Maxwell equations in a medium can be presented in 4-dimensional complex matrix form with the use of two sets of matrices, exploited above: