Products of symplectic normal matrices

A matrix A∈M2n(C) is symplectic if AT[0In−In0]A=[0In−In0]. We show that every symplectic matrix is a product of a symplectic unitary and a symplectic skew-Hermitian matrix. We show that every symplectic matrix is a product of four symplectic skew-Hermitian matrices or a product of four symplectic Hermitian matrices. We give the possible Jordan canonical forms of symplectic matrices which can be written as a product of a symplectic Hermitian and a matrix which is either symplectic Hermitian or symplectic skew-Hermitian.