We consider structures on a Fano plane
which allow a generalisation of Freudenthal’s construction of a norm
and a bilinear multiplication law on an eight-dimensional vector space
canonically
associated to
.
We first determine necessary and sufficient conditions in terms of the incidence geometry
of
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.
