In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree $d>0$ will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type $G_2$, type $F_4$, or simply connected of type $E_6$. We extend the list of cases where the answer is ‘yes’ to all groups of type $G_2$ and some nonsplit groups of type $F_4$ and $E_6$. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.