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

On the Symplectic Structures in Frame Bundles and the Finite Dimension of Basic Symplectic Cohomologies