Homotopical Robustness of Isometries on the Cayley Plane

We compute the effect of the inclusion ${\mathrm{F}}_4\to \mathrm{h}\mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{O}{P}^2)$ on rational homotopy groups by determining the rational homotopy class of the isometry action ${\mathrm{F}}_4\times \mathbb{O}{P}^2\to \mathbb{O}{P}^2$ in terms of a homomorphism between the minimal models of source and target. Although special emphasis is put on the Cayley plane, our results generalise to all simply connected rank $1$-symmetric spaces.