Very recently, Breaz and Cîmpean introduced and examined in Bull. Korean Math. Soc. (2018) the class of so-called weakly tripotent rings as those rings $R$ whose elements satisfy at leat one of the equations $x^3=x$ or $(1-x)^3=1-x$. These rings are generally non-commutative. We here obtain a criterion when the commutative group ring $RG$ is weakly tripotent in terms only of a ring $R$ and of a group $G$ plus their sections. Actually, we also show that these weakly tripotent rings are strongly invo-clean rings in the sense of Danchev in Commun. Korean Math. Soc. (2017). Thereby, our established criterion somewhat strengthens previous results on commutative strongly invo-clean group rings, proved by the present author in Univ. J. Math. Math. Sci. (2018). Moreover, this criterion helps us to construct a commutative strongly invo-clean ring of characteristic $2$ which is not weakly tripotent, thus showing that these two ring classes are different.