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?