Group rings whose symmetric elements are Lie nilpotent

Abstract:

Let $FG$ be the group ring of a group $G$ over a field $F$, with characteristic different from $2$. Let $\ast$ denote the natural involution on $FG$ sending each group element to its inverse. Denote by $(FG)^{+}$ the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided $G$ has no $2$-elements, if $(FG)^{+}$ is Lie nilpotent, then so is $FG$. In this paper, we determine when $(FG)^{+}$ is Lie nilpotent, if $G$ does contain $2$-elements.