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In this paper we study invariant differential operators on manifolds with a given parabolic structure. The model for the parabolic geometry is the quotient of the orthogonal group by a maximal parabolic subgroup corresponding to crossing of the $k$-th simple root of the Dynkin diagram. In particular, invariant differential operators discussed in the paper correspond (in a flat model) to the Dirac operator in several variables.

We prove that the exceptional complex Lie group ${\mathrm{F}}_4$ has a transitive action on the hyperplane section of the complex Cayley plane ${𝕆\mathbb{P}}^2$. 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 $\mathrm{Spin}(9,ℂ)\le {\mathrm{F}}_4$. Moreover, we identify the stabilizer of the ${\mathrm{F}}_4$-action as a parabolic subgroup ${\mathrm{P}}_4$ (with Levi factor ${\mathrm{B}}_3{\mathrm{T}}_1$) of the complex Lie group ${\mathrm{F}}_4$. In the real case we obtain an analogous realization of ${{\mathrm{F}}_4}^{(-20)}/\mathbb{P}$.