Homotopical Robustness of Isometries on the Cayley Plane

We compute the effect of the inclusion $\mathrm{F}_4 \rightarrow \mathrm{hAut}(\mathbb{O}P^2)$ on rational homotopy groups by determining the rational homotopy class of the isometry action $\mathrm{F}_4 \times \mathbb{O}P^2 \rightarrow \mathbb{O}P^2$ 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.