$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion: $$ \SU(N)\ni m = a\, u \, b $$ where $a,b$ are independent diagonal $\SU(N)$-matrices each of which accounts for $N-1$ parameters while $u\in \SU(N)$ is parametrized by the remain $(N-1)^2$ parameters. For instance the matrices $u$ might form a $U(N)$-isomorphic subgroup of $\SU(N)$. Notice that in the case $N=2$ this decomposition reduces to the known Euler's one $a(b)= \exp(i \alpha(\beta) \sigma_3)$ and $U = \exp(i \gamma \sigma_2)$ where $\sigma_j$ is a Pauli matrix and $\alpha,\beta,\gamma$ are the angles.

I found in literature here and there other kinds of generalizations of Euler angles decompositions, however they differ in form from the one above.

Can you point me to some relevant literature? Can you show me a working parametrization of $u$ such that the whole $\SU(N)$ is covered?