Derived categories of the Cayley plane and the coadjoint Grassmannian of type F

For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type $\mathrm{E}_6$, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A general hyperplane section of the Cayley plane is the coadjoint Grassmannian of Dynkin type $\mathrm{F}_4$. We show that the restriction of the Faenzi-Manivel collection to such a hyperplane section gives a full Lefschetz exceptional collection, providing the first example of a full exceptional collection on a homogeneous variety of Dynkin type $\mathrm{F}$. We also describe the residual categories of these Lefschetz collections, confirming conjectures of the second and third named author for the Cayley plane and its hyperplane section. The latter description is based on a general result of independent interest, relating residual categories of a variety and its hyperplane section.