Definition
For a symplectic vector space , a subspace is called isotropic if
Where denotes the symplectic complement
For a symplectic vector space with an isotropic subspace , induces a canonical symplectic form on
Proof
Let for and .
First, is this even well-defined?
Let
Since the vectors in the second and third terms are symplectic complements, and in the fourth is isotropic.
Thus, for , and is well defined.
inherits bilinearity and antisymmetry from , so we just need to check that it is nondegenerate.
Suppose such that for all .
Thus, and .