  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.
