Standard identities for skew-symmetric matrices

Let n be a positive, even integer and let Kn(F) denote the subspace of skew-symmetric matrices of Mn(F), the full matrix algebra with coefficients in a field F. A theorem of Kostant states that Kn(F) satisfies the (2n-2)-fold standard identity s2n-2. In this paper we refine this result by showing that s2n-2 may be written non-trivially as the sum of two polynomial identities of Kn(F).