  In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan
identity, for a sensible theory of quantum mechanics. All but one of the
algebras that satisfy this condition can be described by Hermitian matrices
over the complexes or quaternions. The remaining, exceptional Jordan algebra
can be described by 3x3 Hermitian matrices over the octonions.
  We first review properties of the octonions and the exceptional Jordan
algebra, including our previous work on the octonionic Jordan eigenvalue
problem. We then examine a particular real, noncompact form of the Lie group
E6, which preserves determinants in the exceptional Jordan algebra.
  Finally, we describe a possible symmetry-breaking scenario within E6: first
choose one of the octonionic directions to be special, then choose one of the
2x2 submatrices inside the 3x3 matrices to be special. Making only these two
choices, we are able to describe many properties of leptons in a natural way.
We further speculate on the ways in which quarks might be similarly encoded.
