  Using octonions and the triality property of Spin(8), we find explicit
formulae for the Lie brackets of the exceptional simple real Lie algebras
$\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry
groups of the Cayley projective plane and the Cayley hyperbolic plane. As an
application, we classify polar actions on the Cayley hyperbolic plane which
leave a totally geodesic subspace invariant.
