4 dimensions when a preferred octonionic unit is chosen.

The quaternionic Dirac equation discussed in Section \[QUATERNIONIC\] describes 1 massive particle with 2 spin states, 1 massless particle with only 1 helicity, and their antiparticles. We identify this set of particles with a generation of leptons.

Furthermore, as can be seen from Figure \[Omult\], there is room in the octonions for exactly 3 such quaternionic descriptions which have only their complex part in common, corresponding to replacing $k$ in turn by $i$ and $j$. We identify these 3 quaternionic spaces as describing 3 generations of leptons.

There is, however, one additional massless particle/antiparticle pair, given by (\[Oz\]). Being purely complex, it does not belong to any generation, and it has the opposite helicity from the other massless particles. We do not currently have a physical interpretation for this additional particle; if this theory is to correspond to nature, then this additional particle must for some reason not interact much with anything else.

Note that the mass appears in this theory as an overall scale, which can be thought of as the length scale associated with the corresponding quaternionic direction. In particular, antiparticles must have the same mass as the corresponding 