Symplectic slice for subgroup actions

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