Generating pairs in $\mathfrak g_2$ related by an involution

Let $\mathbb O$ be the real octonion division algebra, let $\mathfrak g_2=\operatorname{Der}(\mathbb O)$, and fix a quaternionic subalgebra $H\subset\mathbb O$. We ask when an element $Z$ preserving $H$ and its conjugate $TZT^{-1}$ generate $\mathfrak g_2$, where $T$ is a prescribed involutive octon