space) in 10 dimensions, which can be nicely described in terms of 2-component spinors over the octonions ${{\mathbb O}}$, the only other normed division algebra besides ${{\mathbb R}}$, ${{\mathbb C}}$, and ${{\mathbb H}}$. Solutions of this equation are automatically quaternionic, and thus lend themselves to the preceding quaternionic description.

The final, and most important, ingredient in our approach is the the dimensional reduction scheme introduced in [@Dim]. In Sections \[CHOOSING\] and \[SPIN\] we describe how the choice of a preferred octonionic unit, or equivalently of a preferred complex subalgebra ${{\mathbb C}}\subset{{\mathbb O}}$, naturally reduces 10 spacetime dimensions to 4, and further allows us to use the standard representation of the Lorentz group $SO(3,1)$ as $SL(2,{{\mathbb C}})\subset SL(2,{{\mathbb O}})$. Putting this all together, we show in Section \[PARTICLES\] that the quaternionic spin/helicity eigenstates correspond precisely to the particle spectrum of a generation of leptons, consisting of 1 massive and 1 massless particle and their antiparticles.

In Section \[SPINOP\], we discuss the remarkable fact that the quaternionic spin eigenstates are in fact simultaneous eigenstates of all 3 spin operators, although the other two eigenvalues are not real. Finally, in Section \[DISCUSSION\] we discuss our results, in particular that, in a natural sense, 