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 $(a+b)^{p^s}=a^{p^s}+b^{p^s}$ if and only if $l\le p^s-p^{s-1}+1$. It is also proved that, under the above conditions on the Lie class of nilpotency, the multiplicative Frobenius relation $(a\cdot b)^{p^s}=a^{p^s}\cdot b^{p^s}$ holds.