Antisymmetric Elements in Group Rings II

Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\vp : RG\to RG$ denote the $R$-linear extension of an involution $\vp$ defined on $G$. An element $x$ in $RG$ is said to be $\vp$-antisymmetric if $\vp (x) = -x$. A characterization is given of when the $\vp$-antisymmetric elements of $RG$ commute. This is a completion of earlier work.