  In this paper we examine the phase space structure of a noncanonical
formulation of 4-dimensional gravity referred to as the Instanton
representation of Plebanski gravity (IRPG). The typical Hamiltonian
(symplectic) approach leads to an obstruction to the definition of a symplectic
structure on the full phase space of the IRPG. We circumvent this obstruction,
using the Lagrange equations of motion, to find the appropriate generalization
of the Poisson bracket. It is shown that the IRPG does not support a Poisson
bracket except on the vector constraint surface. Yet there exists a fundamental
bilinear operation on its phase space which produces the correct equations of
motion and induces the correct transformation properties of the basic fields.
This bilinear operation is known as the almost-Poisson bracket, which fails to
satisfy the Jacobi identity and in this case also the condition of
antisymmetry. We place these results into the overall context of nonsymplectic
systems.
