Specific features of cosquares of special matrices in indefinite metric spaces

As is known, the cosquare of an arbitrary nonsingular Hermitian matrix is the identity matrix, and the cosquare of an arbitrary unitary matrix is again unitary. In a complex linear space with the symplectic metric, symplectic and skew-Hamiltonian matrices are the counterparts of unitary and Hermitian matrices, respectively. Specific features of cosquares for these two matrix classes are indicated.