Quaternionic Reflections and Lie Generation in $G_2$

Let $\mathbb O$ be the real octonion division algebra and $\mathfrak g_2=\operatorname{Der}(\mathbb O)$. A quaternionic reflection $T\in G_2=\operatorname{Aut}(\mathbb O)$ fixes a quaternionic subalgebra pointwise and negates its orthogonal complement. For the associated compact symmetric pair $\mat