$G_2$ as the group of Isometries of a Projective Space

Question

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?

Answer 1

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 $G_2$) clafifies 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.