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

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