Metrics or partial metrics on the octonion linear vector space

Determine which metric or partial metric structures on the eight-dimensional octonion linear vector space are compatible with the multiplicative and associative structure encoded by the Fano plane representation of octonion units.

Background

The paper emphasizes that the octonion units’ multiplicative and associative relations are encoded by lines in the Fano diagram, a projective plane. This projective structure complicates the selection or definition of metric or partial metric structures for the octonion linear vector space.

The author explicitly notes unanswered questions regarding which metric frameworks are appropriate, pointing to an unresolved issue in reconciling the projective incidence structure with a metric or partially metric geometry on the octonion space.