Group algebras whose symmetric and skew elements are Lie solvable

Let $FG$ be the group algebra of a group $G$ without $2$-elements over a field $F$ of characteristic $p\neq 2$ endowed with the canonical involution induced from the map $g\mapsto g^{-1}$, $g\in G$. Let $(FG)^-$ and $(FG)^+$ be the sets of skew and symmetric elements of $FG$, respectively, and let $P$ denote the set of $p$-elements of $G$ (with $P=1$ if $p=0$). In the present paper we prove that if either $P$ is finite or $G$ is non-torsion and $(FG)^-$ or $(FG)^+$ is Lie solvable, then $FG$ is Lie solvable. The remaining cases are also settled upon small restrictions.