Hyperplane section ${\mathbb{O}\mathbb{P}}_0^2$ of the complex Cayley plane as the homogeneous space ${\mathrm{F}}_4\mathrm{/}{\mathrm{P}}_4$

We prove that the exceptional complex Lie group $F_4$ has a transitive action on the hyperplane section of the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of $\text{Spin}(9,\text{C})\le F_4$. Moreover, we identify the stabilizer of the $F_4$-action as a parabolic subgroup $P_4$ (with Levi factor $B_3{T}_1$) of the complex Lie group $F_4$. In the real case we obtain an analogous realization of $F_4^{(-20)}\mathrm{/}{P}_4$.