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We show that the conformally invariant Yamabe operator on a complex conformal manifold can be constructed as a first BGG operator by inducing from certain infinite-dimensional representation.

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 ${{\mathrm{F}}_4}^{(-20)}/\mathbb{P}$.