I'm reading Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society 39 (2): 145–205 regarding $\mathrm{Spin}(8)$ and trialities. My question is about this sentence:
The bigger group $\mathrm{Spin}(8)$ acts as automorphisms of the triality that gives the octonions, and it does so in an interesting way. Given any element $g \in \mathrm{Spin}(8)$, there exists unique elements $g_\pm \in \mathrm{Spin}(8)$ such that $$t(g(v_1), g_+(v_2), g_-(v_3)) = t(v_1,v_2,v_3) $$ for all $v_1 \in V_8, v_2 \in S^+_8,$ and $v_3 \in S^-_8$. Moreover, the maps $$\alpha_\pm : g \to g_\pm $$ are outer automorphisms of $\mathrm{Spin}(8)$.
As my understanding there are not unique elements, since the $g$ acts by $\mathrm{Ad}$, $g\pm$ as the representation in $\Delta\pm$ for example for $g=-1$ one can take $\pm1$ and $\pm1$ to be $g_+$ and $g_-$ or one can take $g_+=\omega$ and $g_-=\omega$, where $\omega$ is the volume form. Moreover $\alpha_{\pm}=\mathrm{Id}$ satisfies the condition so this does not give the outer automorphisms.My question is: Why does Baez say they are unique?
