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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 14th 2012

added to G2 the definition of $G_2$ as the subgroup of $GL(7)$ that preserves the associative 3-form.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 20th 2016
• (edited Jan 20th 2016)

Added (here) the characterization of the subgroups of $G_2 = Aut(\mathbb{O})$ that stabilize and that fix, respectively, the quaternions $\mathbb{H} \hookrightarrow\mathbb{O}$:

$\array{ 1 &=& 1 \\ \downarrow && \downarrow \\ Fix_{G_2}(\mathbb{H}) & \simeq & SU(2) \\ \downarrow && \downarrow \\ Stab_{G_2}(\mathbb{H}) &= & Stab_{G_2}(\mathbb{H}) \\ \downarrow && \downarrow \\ Aut(\mathbb{H}) &\simeq& SO(3) \\ \downarrow && \downarrow \\ 1 &=& 1 }$
• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJan 20th 2016

I was wondering if your middle group had another name. Is this saying it is $SO(4)$?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 20th 2016
• (edited Jan 20th 2016)

Yes, true. Thanks. The source which I had cited also said this, but I forgot to include it. Done now.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 20th 2016

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.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeOct 30th 2017
• (edited Oct 30th 2017)

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

1. Even a simple dimension count reveals that it cannot possible be the six-sphere.

Anonymous

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 12th 2019

14-8=6

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 13th 2019

I have reverted the edit in revision 31 by “Anonymous” above and put in a link to G2/SU(3) is the 6-sphere

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeApr 14th 2019

Under Orientation, did you mean to write $SO(7)$ instead of $SL(7)$?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 14th 2019

Thanks, fixed now.

(Pointer to a reference is missing here, but I don’t have time for it right now.)