  As a direct continuation of K. Zwart, arXiv:2304.02648, which is built on the
work of M. M\"uger and L. Tuset, we reduce the Mathieu conjecture, formulated
by O. Mathieu in 1997, for $Sp(N)$ and $G_2$ to a conjecture involving
functions over $\mathbb{R}^n\times (S^1)^m$ with $n,m\in\mathbb{N}_0$. The
proofs rely on Euler-style parametrizations of these groups, a specific version
of the $KAK$ decomposition, which we discuss and prove.
