  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.
