One other parameterization of SU(4) group

We propose a special decomposition of the Lie su(4) algebra into the direct sum of orthogonal subspaces, su(4)=k^\,, with k=su(2)(2) and a triplet of 3-dimensional Abelian subalgebras (a, a^, t)\,, such that the exponential mapping of a neighbourhood of the 0∈ su(4) into a neighbourhood of the identity of the Lie group provides the following factorization of an element of $SU(4)$ \[ g = k\,a\,t\,, \] where k ∈ exp(k) = SU(2)× SU(2) ⊂ SU(4)\,, the diagonal matrix t stands for an element from the maximal torus T³=exp(t), and the factor a=exp(a)exp(a⁾ corresponds to a point in the double coset SU(2)× SU(2) SU(4)/T³. Analyzing the uniqueness of the inverse of the above exponential mappings, we establish a logarithmic coordinate chart of the $SU(4)$ group manifold comprising 6 coordinates on the embedded manifold SU(2)× SU(2) ⊂ SU(4) and 9 coordinates on three copies of the regular octahedron with the edge length 2π√2\,.