Each 2n-by-2n complex symplectic matrix is a product of n + 1 commutators of J-symmetries

A 2n×2n complex matrix A is symplectic if A⊤[0I−I0]A=[0I−I0]. If A is symplectic and rank(A−I)=1, then it is called a J-symmetry. For each n, we prove that every 2n×2n symplectic matrix M is a product of n+1 commutators of J-symmetries and this number cannot be smaller for some M.