Perhaps this will help. Please note that it's written by Joan Baez' alter (and secret) identity.

Author: John C. Baez
Comments: 56 pages LaTeX, 11 Postscript Figures, some small corrections
Subj-class: Rings and Algebras; Representation Theory; Quantum Algebra; History and Overview

    The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.