  We present a construction (and classification) of certain invariant 2-forms
on the real symplectic group. They are used to define a symplectic form on the
quotient by a maximal torus and to "lift" a symplectic structure from a
symplectic manifold to the bundle of frames. This is a by-product of a failed
attempt to prove certain finiteness theorems for basic symplectic cohomologies.
In the last part of the paper we include a valid proof.
