Abstract

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.

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