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 etale 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 G₂, type F₄, or simply connected of type E₆. We extend the list of cases where the answer is ``yes" to all groups of type G₂ and some nonsplit groups of type F₄ and E₆. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.