in order to avoid conflict with our subsequent conventions for the octonions, we will instead label our quaternionic basis $\ell$, $k$, $\ell k$. The imaginary unit $\ell$ will play the role of the complex unit $i$, and, as we will see later, $k$ will label this particular quaternionic subalgebra of the octonions. In terms of the Cayley-Dickson process [@Dickson; @Schafer], we have $$\label{Cayley}
{{\mathbb H}}= {{\mathbb C}}+ {{\mathbb C}}k = ({{\mathbb R}}+ {{\mathbb R}}\ell) + ({{\mathbb R}}+ {{\mathbb R}}\ell)k$$

As vector spaces, ${{\mathbb H}}= {{\mathbb C}}^2$, which allows us to identify ${{\mathbb H}}^2$ with ${{\mathbb C}}^4$ in several different ways. We choose the identification $$\label{IdentifyC}
\pmatrix{A\cr B\cr C\cr D\cr}
  \longleftrightarrow
  \pmatrix{C - k B\cr \noalign{\smallskip} D + k A\cr}$$ with $A,B,C,D\in{{\mathbb C}}$. Equivalently, we can write this identification in terms of the Weyl (Penrose) spinors $\theta$ and $\eta$ as $$\label{IdentifyK}
\Psi = 
\pmatrix{\theta\cr \noalign{\smallskip} \eta\cr}
  \longleftrightarrow
  \eta + \sigma_k \theta$$ where we have introduced the generalized Pauli matrix $$\sigma_k = \pmatrix{0& -k\cr \noalign{\smallskip} k& 0\cr}$$

Since (\[IdentifyK\]) is clearly a vector space isomorphism, there is also an isomorphism relating the linear maps on these spaces. We can use the induced isomorphism to rewrite the Dirac equation (\[DiracI\]) in 2-component quaternionic 