  We consider constrained Hamiltonian systems in the framework of Dirac's
theory. We show that the Jacobi identity results from imposing that the
constraints are Casimir invariants, regardless of the fact that the matrix of
Poisson brackets between constraints is invertible or not. We point out that
the proof we provide ensures the validity of the Jacobi identity everywhere in
phase space, and not just on the surface defined by the constraints. Two
examples are considered: A finite dimensional system with an odd number of
constraints, and the Vlasov-Poisson reduction from Vlasov-Maxwell equations.
