Hyperplane section $\mathbb{OP}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$

We prove that the exceptional complex Lie group $F_4$ has a transitive action on the hyperplane section of the complex Cayley plane $\mathbb{OP}^2$. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of $\Spin(9,\C) \leq F_4$. Moreover, we identify the stabilizer of the $F_4$-action as a parabolic subgroup $P_4$ (with Levi factor $B_3T_1$) of the complex Lie group $F_4$. In the real case we obtain an analogous realization of $F_4^{(-20)}/P_4$.