  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.
