  We give a streamlined account of $2$-spinors, up to and including the Dirac
equation, using little more than the resources of linear algebra. We prove that
the Dirac bundle is isomorphic to the associated bundles
$\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} S$ and
$\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} \bar{S}$. A solution of the
Dirac equation determines a pair of conjugate $2$-spinor fields over the mass
shell $X_m$.
