q_\mu \sigma^\mu$$ defines the [*octonionic Pauli matrices*]{} $\sigma_\mu$. The matrices $\sigma_a$, with $a=1,...,9$, are the natural generalization of the ordinary Pauli matrices to the octonions, and $\sigma_0=I$. In analogy with our treatment of the complex case, we have $$\label{OctoI}
\gamma_0 \gamma_\mu
    = \pmatrix{-{\widetilde}\sigma_\mu& 0\cr \noalign{\smallskip} 0& \sigma_\mu\cr}$$

For completeness, we record some useful relationships. The adjoint ${\overline}\Psi$ of the Majorana spinor $\Psi$ is given as usual by $${\overline}\Psi = \Psi^\dagger \gamma_0$$ since $$\gamma_\mu^\dagger \gamma_0^\dagger = \gamma_0 \gamma_\mu$$ Given a Majorana spinor $\Psi=\pmatrix{\psi\cr\noalign{\smallskip}\chi}$, we can construct a real vector [^5] $$q^\mu[\Psi] = {{\rm Re}}(\Psi^\dagger\gamma^0\gamma^\mu\Psi)$$ corresponding in traditional language to ${\overline}\Psi\gamma^\mu\Psi$. We can further identify this with a $2\times2$ matrix $Q[\Psi]$ as in (\[MatDef\]) above. Direct computation using the cyclic property of the trace, e.g. for octonionic columns $\Psi_1$, $\Psi_2$, and octonionic matrices $\gamma$ $${{\rm Re}}(\Psi_1^\dagger \gamma \Psi_2^{\phantom\dagger})
  = {{\rm Re}}\left({{\rm tr\,}}(\Psi_1^\dagger \gamma \Psi_2^{\phantom\dagger})\right)
  = {{\rm Re}}\left({{\rm tr\,}}(\Psi_2^{\phantom\dagger} \Psi_1^\dagger \gamma)\right)$$ shows that [@FairlieI; @FairlieII] $$\label{MSq}
Q[\Psi] = 2 \, \psi\psi^\dagger - 2 \, {\widetilde}{\chi\chi^\dagger}$$

Octonionic Dirac Equation
-------------------------

The momentum-space massless Dirac equation (Weyl equation) in 10 dimensions can be written in the form $$\label{Weyl}
\gamma_0\gamma_\mu \, p^\mu \, \Psi = 0$$ Choosing $\Psi=\pmatrix{\psi\cr\noalign{\smallskip}0}$ to be a Majorana-Weyl spinor, and using (\[OctoI\]) 