# analogues of Cayley plane as homogenous spaces

The Cayley projective plane $\mathbb{OP}^2$ can be defined as a homogenous space $\mathrm{F_4/Spin(9)}$, where $\mathrm{F_4}$ is the compact exceptional simple Lie group. The other possible approach is to define it as equivalence classes of (special) triples of octonions, define a Riemannian metric on it and prove that the group of isometries is compact Lie group of type $\mathrm{F_4}$. This is done in this article. There the authors study also other spaces similar to $\mathbb{OP}^2$, namely they consider the octionionic hyperbolic plane,the octonionic projective plane with indefinite signature and analogue of projective plane made up from split octonions.

My question is: What are the "homogeneous presentations" of these spaces?

There are three real forms of the complex simple Lie algebra of type $F_4$. Each such real form will have (real) Lie subalgebras whose complexification is $\mathfrak{so}(9;\mathbb{C})$. Each such pair gives rise to homogeneous 16-dimensional pseudoriemannian manifolds.
• The authors of the paper prove that each of the four manifolds is homogeneous and symmetric. I do not know the theory of symmetric spaces and the wikipedia pages are a bit confusing for me. Anyway, if I understand the classification list on wikipedia, I have only two spaces - one homogeneous under compact $F_4$ and the other under $F_4^{(-20)}$ (the hyperbolic plane). I guess the other two could be find among pseudo-Riemannian symmetric in the table here: en.wikipedia.org/wiki/Symmetric_space#Tables By the way, the paper was published in a polished form in DGA 27 (2009). Jan 4 '11 at 23:34
• There is no classification of pseudo-riemannian symmetric spaces, I'm afraid, which is why the wikipedia page restricts itself to symmetric pairs $(\mathfrak{g},\mathfrak{h})$ where $\mathfrak{g}$ is simple. Indeed, as you point out, there are only two riemannian examples among the cases at hand: the Cayley plane $F_4/\mathrm{Spin}(9)$ and its noncompact dual. Both of these have $\mathrm{Spin}(9) \subset \mathrm{SO}(16)$ isotropy representation, but the transvection group is different: the compact $F_4$ in one case and the maximally split $F_4$ in the other. (TBC) Jan 5 '11 at 0:30