Polynomial identities for matrices symmetric with respect to the symplectic involution

Let m>1 be a positive integer, F be a field, and let H2m(F,s) denote the subspace of M2m(F) of matrices symmetric with respect to the symplectic involution. We show that H2m(F,s) satisfies a multilinear identity of degree 4m−3, and via this identity we obtain a refinement of a theorem of Rowen concerning s4m−2, a so-called “standard” polynomial identity for H2m(F,s).