Incidence geometry of the Fano plane and Freudenthal's ansatz for the construction of (split) octonions

Abstract

In this article we consider structures on a Fano plane $\mathcal{F}$ which allow a generalisation of Freudenthal's construction of a norm and a bilinear multiplication law on an eight-dimensional vector space ${\mathbb{O}}_{\mathcal{F}}$ canonically associated to $\mathcal{F}$. We first determine necessary and sufficient conditions in terms of the incidence geometry of $\mathcal{F}$ for these structures to give rise to division composition algebras, and classify the corresponding structures using a logarithmic version of the multiplication. We then show how these results can be used to deduce analogous results in the split composition algebra case.