Lie derived length and involutions in group algebras

Let $G$ be a group such that the set of $p$-elements of $G$ forms a finite nonabelian subgroup, where $p$ is an odd prime, and let $F$ be a field of characteristic $p$. In this paper we prove that the lower bound of the Lie derived length of the group algebra $FG$ given by Shalev in [11] is also a lower bound for the Lie derived length of the set of symmetric elements of $FG$ for every involution which is linear extension of an involutive anti-automorphism of $G$. Furthermore, we provide counterexamples to the interesting cases which are not covered by the main theorem.