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.