On the Mathieu Conjecture for Sp (N) and G₂

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 G₂ to a conjecture involving functions over Rⁿ (S¹) ᵐ with n, m₀. The proofs rely on Euler-style parametrizations of these groups, a specific version of the KAK decomposition, which we discuss and prove.