One other parameterization of SU(4) group

We propose a special decomposition of the Lie \(\mathfrak{su}(4)\) algebra into the direct sum of orthogonal subspaces, \(\mathfrak{su}(4)=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{a}^\prime\oplus\mathfrak{t}\,,\) with \(\mathfrak{k}=\mathfrak{su}(2)\oplus\mathfrak{su}(2)\) and a triplet of 3-dimensional Abelian subalgebras \((\mathfrak{a}, \mathfrak{a}^{\prime}, \mathfrak{t})\,,\) such that the exponential mapping of a neighbourhood of the \(0\in \mathfrak{su}(4)\) 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 \(k \in \exp{(\mathfrak{k})} = SU(2)\times SU(2) \subset SU(4)\,,\) the diagonal matrix \(t\) stands for an element from the maximal torus \(T^3=\exp{(\mathfrak{t})},\) and the factor \(a=\exp{(\mathfrak{a})}\exp{(\mathfrak{a}^\prime)}\) corresponds to a point in the double coset \(SU(2)\times SU(2)\backslash SU(4)/T^3.\) 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 \( SU(2)\times SU(2) \subset SU(4)\) and 9 coordinates on three copies of the regular octahedron with the edge length \(2\pi\sqrt{2}\,\).