Identities on Lie or Jordan-group-graded associative algebras

Let G be a finite group. Bahturin, Giambruno and Riley proved that if A is an (associatively) G-graded associative algebra such that the homogeneous component A1 satisfies a polynomial identity of degree d, then the entire algebra A satisfies a polynomial identity with degree bounded above by an explicit function of d and |G|. We extend this result to include associative algebras A that are either Lie or Jordan-G-graded. We deduce the following sharpening of a well-known theorem of Amitsur: if the invariant (respectively, skew-invariant) subspace of a Jordan (respectively, Lie) involution on A satisfies a polynomial identity of degree d, then the entire algebra A satisfies a polynomial identity of degree bounded above by an explicit function of d.