Octonions, triality, the exceptional Lie algebra F4, and polar actions on the Cayley hyperbolic plane

TL;DR

This paper derives explicit formulas for the Lie brackets of the exceptional Lie algebras F4 and F4* using octonions and triality, and classifies certain symmetry actions on the Cayley hyperbolic plane.

Contribution

It provides explicit Lie bracket formulas for F4 and F4* using octonions and triality, and classifies polar actions on the Cayley hyperbolic plane.

Findings

Explicit formulas for Lie brackets of F4 and F4*

Classification of polar actions on the Cayley hyperbolic plane

Identification of symmetries leaving geodesic subspaces invariant

Abstract

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.