Totaro's Question on Zero-Cycles on G2, F4 and E6 Torsors

In a 2004 paper, Totaro asked whether a G-torsor X that hasa zero-cycle of degree d > 0 will necessarily have a closedétale point of degree dividing d, where G is a connectedalgebraic group. This question is closely related to severalconjectures regarding exceptional algebraic groups. Totaro gavea positive answer to his question in the following cases: Gsimple, split, and of type G₂, type F₄, or simply connectedof type E₆. We extend the list of cases where the answer is‘yes’ to all groups of type G₂ and some nonsplitgroups of type F₄ and E₆. No assumption on the characteristicof the base field is made. The key tool is a lemma regardinglinkage of Pfister forms.