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We prove that the exceptional complex Lie group $F_4$ has a transitive actionon the hyperplane section of the complex Cayley plane $\mathbb{OP}^2$. Ourproof is direct and constructive. We use an explicit realization of the vectorand spin actions of $\Spin(9,\C) \leq F_4$. Moreover, we identify thestabilizer 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 ananalogous realization of $F_4^{(-20)}/P_4$.

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