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It seems like in the classification of simple complex lie algebras, every lie algebra corresponds to the group of isometries of a projective space. SO(n+1) is the group of isometries on $RP^n$, SU(n+1) is the isometries of $CP^n$, and SP(n+1) is the isometries of $HP^n$.

John Baez explains in his course on the octonions that the exceptional lie groups are the isometries groups for projective spaces built out of the octonions, as seen in the Magic Square of Lie Algebras1

$G_2$ is the only exceptional lie group left out of this description, and is usually described as the group of automorphisms of the Octonians, which is nice, but following the pattern it seems it should be the group of isometries of some manifold as well. Is it known what this manifold would be?

Cheyne
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  • See the last entry of this table. It is not a projective space though. – pregunton Nov 25 '20 at 11:53
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    This is relevant. – Tyrone Nov 26 '20 at 16:16
  • Thanks, Tyrone. Pregunton, it seems I had the exact same questions as you 5 years later. Though I'm confused about thinking of these as symmetries of Symmetric spaces. This seems to be different than isometries of a manifold? At least for the usual lie groups, it doesn't describe projective spaces. It is still interesting that the classification of the two objects are related though. – Cheyne Nov 26 '20 at 21:10
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    Well, symmetric spaces are manifolds, and the group labeled as $G$ in the table is actually (the identity component of) their isometry group, see e.g. this question, or the first sentence here in the same page. It's not obvious, but the projective spaces associated to the classical Lie groups do appear in the table: they are the $p=1$ cases of AIII, BDI and CII. – pregunton Nov 26 '20 at 22:57
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    The G space thus has $G_2$ as its isometry group, so it is technically an answer to your question, but I don't really consider it a satisfying answer for several reasons: it doesn't have the same properties as the other generalized projective spaces (with respect to tangent spaces, etc.), it doesn't seem to be related to the usual magic square construction in any way, and to me it feels more like a generalization of the AI and CI cases. – pregunton Nov 26 '20 at 22:58
  • From the point of view of generalized flag manifolds, the closest analogy is probably the following: Consider the $5$-dimensional null complex quadric space $\mathcal N$ in $\Bbb C \Bbb P^6$, in turn viewed as the projectivization of the vector space of purely imaginary complex octonions. Octonionic multiplication induces a maximally nonintegrable (complex) $2$-plane distribution $\mathbf D$ on $\mathcal N$, and the subgroup of $PSL(7, \Bbb C)$ fixing $(\mathcal N, \mathbf D)$ is the complex simplex Lie group $G_2$; in particular, $G_2$ acts transitively on $\mathcal N$. – Travis Willse Jul 22 '22 at 22:59

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Too long for a comment, but not a full answer:

There is a famous realization as $G_2$ as the symmetry group of 'a ball rolling over another ball with 3 times its radius'.

I don't really know what that means, but whenever you invent a sensible parametrization of all possible configurations of the two balls it is not hard to convince yourself that his thing has the structure of a manifold. Perhaps this manifold is the thing that has $G_2$ symmetry. On the otherhand, this is just two balls touching. If somehow the notion of rolling plays a more serious role it is less obvious if and how the story can be reformulated as a manifold.

But a good starting point would be to google '$G_2$ rolling ball' or similar and see what that turns up.

EDIT: this quote from Wikipedia (the page on $G_2$) clarifies it quite a lot:

In 1893, Élie Cartan published a note describing an open set in $\mathbb{C}^5$ equipped with a 2-dimensional distribution—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra $\mathfrak{g}_{2}$ appears as the infinitesimal symmetries.[2] In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.

Vincent
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  • This is very interesting! Doing some more googling, I found this paper by Baez where he views G2 as the symmetries of the 'spinorial ball rolling on a projective plane'. So it does seem to somehow relate back to projective space – Cheyne Nov 25 '20 at 17:00
  • Reading more, it seems like G2 may be related to the symmetries of the split octonian projective space? – Cheyne Nov 25 '20 at 17:18
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    @Cheyne The isometries of the (split) octonionic plane are respectively compact (resp., split) groups of type $F_4$, but the compact (resp., split) form of the group $G_2$ can be viewed as the subgroup of collineations of the (split) octonionic plane fixing a particular collection of points. – Travis Willse Jul 22 '22 at 22:51