Symplectic slice for subgroup actions

Given a symplectic manifold \((M,\omega)\) endowed with a proper Hamiltonian action of a Lie group \(G\), we consider the action induced by a Lie subgroup \(H\) of \(G\). We propose a construction for two compatible Witt-Artin decompositions of the tangent space of \(M\), one relative to the \(G\)-action and one relative to the \(H\)-action. In particular, we provide an explicit relation between the respective symplectic slices.