Does every element of $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilize a quaternion subalgebra?

Question

Has anyone heard of this result before?

Let $\mathbb{O}$ denote the octonions and let $G_2$ denote its automorphism group (i.e. the 14 dimensional subgroup of $SO(7)$). Then any element of $G_2$ stabilizes at least one quaternion subalgebra of $\mathbb{O}$.

I couldn't find it in Conway's book.

Thanks!