the context of 4-component [*Dirac spinors*]{}. The 4 degrees of freedom correspond to the choice of spin (up or down) and the choice of particle or antiparticle. Similarly, 2-component [*Penrose spinors*]{}, which can be thought of as the square roots of null vectors, correspond to massless objects, such as photons. In Section \[COMPLEX\] we set the stage by reviewing these standard properties of the chiral description of the momentum-space Dirac equation.

Penrose spinors are usually thought of as Weyl projections of Dirac spinors; a Dirac spinor contains twice the information of a single Penrose spinor. As an alternative to doubling the number of (complex) components, however, we double the dimension of the underlying division algebra, from the complex numbers ${{\mathbb C}}$ to the quaternions ${{\mathbb H}}$. The anticommutativity of the quaternions then enables us to package two complex representations of opposite chirality into the (now quaternionic) 2-component formalism. In Section \[QUATERNIONIC\] we show how to replace the usual 4-component complex Dirac description with an equivalent 2-component quaternionic Penrose description, and further discuss how this puts the massive and massless Dirac equations on an equal footing.

We then consider in Section \[OCTONIONIC\] the massless Dirac equation on Majorana-Weyl spinors (in momentum 