---
abstract: 'We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic spinors, leading to a formalism which treats both massive and massless particles on an equal footing. The resulting unified description has the correct particle spectrum to be a generation of leptons, with the correct number of spin/helicity states. Furthermore, precisely three such generations naturally combine into an octonionic description of the 10-dimensional massless Dirac equation, as previously discussed in [@Dim].'
author:
- |
    Tevian Dray\
    [*Department of Mathematics, Oregon State University, Corvallis, OR 97331*]{}\
    [tevian[@]{}math.orst.edu]{}\
- |
    Corinne A. Manogue\
    [*Department of Physics, Oregon State University, Corvallis, OR 97331*]{}\
    [corinne[@]{}physics.orst.edu]{}
date: '1 December 1998 (revised 30 September 1999)'
title: '**QUATERNIONIC SPIN [^1]** '
---

**INTRODUCTION**
================

We recently outlined a new dimensional reduction scheme [@Dim]. We show here in detail that applying this mechanism to the 10-dimensional massless Dirac equation on Majorana-Weyl spinors leads to a quaternionic description of the full 4-dimensional (free) Dirac equation which treats both massive and massless particles on an equal footing. Furthermore, there are naturally 3 such descriptions, each of which corresponds to a generation of leptons with the correct number of spin/helicity states.

The massive Dirac equation is usually formulated in 