Geodesic
Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 535-549.

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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 ${\mathrm{F}_4}^{(-20)}/\P$.
Classification : 14M17, 32M12
Keywords: Cayley plane; octonionic contact structure; twistor fibration; parabolic geometry; Severi varieties; hyperplane section; exceptional geometry 
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     title = {Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex {Cayley} plane as the homogeneous space $\mathrm{F_4/P_4}$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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Pazourek, Karel; Tuček, Vít; Franek, Peter. Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 535-549. http://geodesic.mathdoc.fr/item/CMUC_2011__52_4_a5/