One other parameterization of SU(4) group

We propose a special decomposition of the Lie $\mathfrak{s}\mathfrak{u}(4)$ algebra into the direct sum of orthogonal subspaces, $\mathfrak{s}\mathfrak{u}(4)=\mathfrak{k}\oplus \mathfrak{a}\oplus {\mathfrak{a}}^{\mathrm{\prime }}\oplus \mathfrak{t},$ with $\mathfrak{k}=\mathfrak{s}\mathfrak{u}(2)\oplus \mathfrak{s}\mathfrak{u}(2)$ and a triplet of 3-dimensional Abelian subalgebras $(\mathfrak{a},{\mathfrak{a}}^{\mathrm{\prime }},\mathfrak{t}),$ such that the exponential mapping of a neighbourhood of the $0\in \mathfrak{s}\mathfrak{u}(4)$ into a neighbourhood of the identity of the Lie group provides the following factorization of an element of $SU(4)$ $$g=kat,$$ 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 $a=\exp (\mathfrak{a})\exp ({\mathfrak{a}}^{\mathrm{\prime }})$ corresponds to a point in the double coset $SU(2)\times SU(2)\mathrm{\backslash }SU(4)\mathrm{/}{T}^3\mathrm{.}$ 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 $2\pi \sqrt{2}$.