  According to the Weinstein splitting theorem, any Poisson manifold is
locally, near any given point, a product of a symplectic manifold with another
Poisson manifold whose Poisson structure vanishes at the point. Similar
splitting results are known e.g. for Lie algebroids, Dirac structures and
generalized complex structures. In this paper, we develop a novel approach
towards these results that leads to various generalizations, including their
equivariant versions as well as their formulations in new contexts.
