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

Abstract

summary:We prove that the exceptional complex Lie group ${\mathrm{F}_4}$ has a transitive action on the hyperplane section of the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of ${\mathrm{Spin}}(9,\mathbb{C})\leq {\mathrm{F}_4}$. Moreover, we identify the stabilizer of the ${\mathrm{F}_4}$-action as a parabolic subgroup ${\mathrm{P}_4}$ (with Levi factor $\mathrm{B_3T_1}$) of the complex Lie group ${\mathrm{F}_4}$. In the real case we obtain an analogous realization of ${\mathrm{F}_4}^{(-20)}/\P$