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added to G2 the definition of $G_2$ as the subgroup of $GL(7)$ that preserves the associative 3-form.
Added (here) the characterization of the subgroups of $G_2 = Aut(\mathbb{O})$ that stabilize and that fix, respectively, the quaternions $\mathbb{H} \hookrightarrow\mathbb{O}$:
$\array{ 1 &=& 1 \\ \downarrow && \downarrow \\ Fix_{G_2}(\mathbb{H}) & \simeq & SU(2) \\ \downarrow && \downarrow \\ Stab_{G_2}(\mathbb{H}) &= & Stab_{G_2}(\mathbb{H}) \\ \downarrow && \downarrow \\ Aut(\mathbb{H}) &\simeq& SO(3) \\ \downarrow && \downarrow \\ 1 &=& 1 }$I was wondering if your middle group had another name. Is this saying it is $SO(4)$?
Yes, true. Thanks. The source which I had cited also said this, but I forgot to include it. Done now.
Added the argument (here) that $dim(G_2) = 14$ and the argument (here) that $Fix_{G_2}(\mathbb{H}) \simeq SU(2)$, both using the statement that “octonionic basic triples” form a torsor over $G_2$, taken from Baez, 4.1.
I added the reference to Basak17, which builds the root space decomposition of the Lie algebra of $G_2$ from a nice description of the octonions
Tathagata Basak, Root space decomposition of $\mathfrak{g}_2$ from octonions, arXiv:1708.02367
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