Frobenius Relations for Associative Lie Nilpotent Algebras

It is proved that any relatively free associative Lie nilpotent algebra of a class l over a field of finite characteristic p satisfies the additive Frobenius relation (a + b)p(s) = a(ps) + bps if and only if l <= p(s) - p(s- 1) + 1. It is also proved that, under the above conditions on the Lie class of nilpotency, the multiplicative Frobenius relation (a center dot b)(ps) = a(ps) center dot b(ps) holds.