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?