Hessian matrix non-decomposition theorem

In his 1983 invited lecture at the International Congress of Mathematics, R oger Brockett proposed to classify finite dimensional estimation algebras. The following problem arises from the first author's classification theory of finite dimensional estimation algebras with maximal rank. Can the Hessia n matrix of a homogeneous polynomial of degree 4 be decomposed in the form Delta(x)Delta(z)(T) where Delta(x) is an anti-symmetric linear matrix (i.e. , entries of Delta(x) are linear in x)? In this short note, we show that th is cannot be true, in other words, the Hessian matrix is nondecomposable in this form.