In this paper we introduce almost Poisson structures on Lie groups which
generalize Poisson structures based on the use of the classical Yang-Baxter identity.
Almost Poisson structures fail to be Poisson structures in the sense that they do
not satisfy the Jacobi identity.In the case of cross products of Lie groups, we show
that an almost Poisson structure can be used to derive a system which is intimately
related to a fundamental Hamiltonian integrable system — the generalized rigid body
equations
