Split-Quaternions and the Dirac Equation

We show that Dirac 4-spinors admit an entirely equivalent formulation in terms of 2-spinors defined over the split-quaternions. In this formalism, a Lorentz transformation is represented as a $2\times 2$ unitary matrix over the split-quaternions. The corresponding Dirac equation is then derived in terms of these 2-spinors. In this framework the $SO(3,2;\mathbf{R})$ symmetry of the Lorentz invariant scalar $\bar{\psi }\psi$ is manifest.