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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
14-8=6
I have reverted the edit in revision 31 by “Anonymous” above and put in a link to G2/SU(3) is the 6-sphere
Under Orientation, did you mean to write $SO(7)$ instead of $SL(7)$?
pointer to
where it is shown that the group of zero-divisors of the sedenions is isomorphic to $G_2$.
By the way, has the observation in Relation to higher prequantum geometry been used anywhere?
No, I am not aware that this point of view has been used anywhere.
Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed one redirect for “G2” from the top and added one for “G2” at the bottom of the page.)
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