have introduced the 2-component quaternionic spinor $$\psi = \sigma_k (\eta + \sigma_k\theta) = \theta + \sigma_k\eta$$ When written out in full, (\[DiracVI\]) takes the form $$\pmatrix{-p^t+p^z& p^x-\ell p^y-km\cr
       \noalign{\smallskip}
       p^x+\ell p^y+km& -p^t-p^z}
  \psi = 0$$ This clearly suggests viewing the mass as an additional spacelike component of a higher-dimensional vector. Furthermore, since the matrix multiplying $\psi$ has determinant zero, this higher-dimensional vector is null. We thus appear to have reduced the massive Dirac equation in 4 dimensions to the massless Dirac, or Weyl, equation in higher dimensions, thus putting the massive and massless cases on an equal footing. This expectation is indeed correct, as we show in the next section in the more general octonionic setting.

**OCTONIONIC FORMALISM** {#OCTONIONIC}
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Octonionic Penrose Spinors
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The [*octonions*]{} ${{\mathbb O}}$ are the nonassociative, noncommutative, normed division algebra over the reals. The octonions are spanned by the identity element $1$ and seven imaginary units, which we label as $\{i,j,k,k\ell,j\ell,i\ell,\ell\}$. Each imaginary unit squares to $-1$ $$i^2 = j^2 = k^2 = ... = \ell^2 = -1$$ and the full multiplication table can be conveniently encoded in the 7-point projective plane, as shown in Figure \[Omult\]; each line is to be thought of 