  We consider the twistor theory of nilconformal harmonic maps from a Riemann
surface into the Cayley plane $\mathbf{O} P^2=F_4/\mathrm{Spin}(9)$. By
exhibiting this symmetric space as a submanifold of the Grassmannian of
$10$-dimensional subspaces of the fundamental representation of $F_4$,
techniques and constructions similar to those used in earlier works on twistor
constructions of nilconformal harmonic maps into classical Grassmannians can
also be applied in this case. The originality of our approach lies on the use
of the classification of nilpotent orbits in Lie algebras as described by D.
Djokovi\'{c}.
