Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let Jn be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G-gradings on Jn where G is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1, where n is the dimension of the vector space V defining Jn. We prove that in this case the algebra Jn is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.