Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?

http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg

The symmetry group of the Fano plane is PSL(2,7), the second-smallest nonabelian simple group. It is also the smallest Hurwitz group, and the group of automorphisms of the Klein quartic.

http://en.wikipedia.org/wiki/PSL(2,7)

I guess I'm wondering if Hurwitz' classification of normed division algebras and Hurwitz' theorem on automorphisms of Riemann surfaces are directly related in some way.

http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) http://en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem

EDIT: Apparently the Fano plane mnemonic is originally due to Freudenthal (1951). Unfortunately I don't read German, nor do I have access to Freudnthal's book to scan for the word "Fano".

http://www.ams.org/mathscinet-getitem?mr=MR13:433a

EDIT: John Baez suggests an idea here (from 2001), but I can't find any follow-up discussion.