In this article we consider structures on a Fano plane ${ \cal F}$ which allow a generalisation of Freudenthal's construction of a norm and a bilinear multiplication law on an eight-dimensional vector space ${ \mathbb O_{ \cal F}}$ canonically associated to ${ \cal F}$. We first determine necessary and sufficient conditions in terms of the incidence geometry of ${ \cal F}$ 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.
Categories: math.CO math.RA
