  We construct a full exceptional collection consisting of vector bundles in
the derived category of coherent sheaves on the so-called Cayley Grassmannian,
the subvariety of the Grassmannian $\mathrm{Gr}(3, 7)$ parameterizing
3-subspaces that are annihilated by a general 4-form. The main step in the
proof of fullness is a construction of two self-dual vector bundles which is
obtained from two operations with quadric bundles that might be interesting in
themselves.
