TITLE: analogues of Cayley plane as homogenous spaces
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: Disclaimer: I have not read the paper, but I think that this what's going on.
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.
This construction certainly accounts for the octonionic projective plane and its noncompact dual, as for the octonionic hyperbolic plane.  I have not worked out the other cases mentioned in the question.