O}}$ is an octonion representing the transverse spatial coordinates. Following [@Sudbery], we define $${\widetilde}{Q} = Q - {{\rm tr\,}}(Q) I$$ Furthermore, since $Q$ satisfies its characteristic polynomial, we have $$-Q{\widetilde}{Q} = -{\widetilde}{Q}Q = -Q^2 + {{\rm tr\,}}(Q) Q = \det(Q) I
  = g_{\mu\nu} q^\mu q^\nu I$$ where $g_{\mu\nu}$ is the Minkowski metric (with signature ($+$ $-$ ... $-$)). We can therefore identify the tilde operation with the metric dual, so that $-{\widetilde}{Q}$ represents the covariant vector field $q_\mu$.

Just as in the complex case (compare (\[CGamma\])), this can be thought of (up to associativity issues) as a Weyl representation of the underlying Clifford algebra ${{\cal C \it l}}(9,1)$ in terms of $4\times4$ gamma matrices of the form $$\label{MatDef}
q^\mu \gamma_\mu
  = \pmatrix{0&Q\cr \noalign{\smallskip} -{\widetilde}{Q}&0\cr}$$ which are now octonionic. It is readily checked that $$\gamma_\mu\gamma_\nu + \gamma_\nu\gamma_\mu = 2\,g_{\mu\nu}$$ as desired.

In this language, a Majorana spinor $\Psi=\pmatrix{\psi\cr\noalign{\smallskip}\chi}$ is a 4-component octonionic column, whose chiral projections are the Majorana-Weyl spinors $\pmatrix{\psi\cr\noalign{\smallskip}0}$ and $\pmatrix{0\cr\noalign{\smallskip}\chi}$, which can be identified with the 2-component octonionic columns $\psi$ and $\chi$, which in turn can be thought of as generalized Penrose spinors. Writing $$\gamma_\mu
  = \pmatrix{0& \sigma_\mu\cr \noalign{\smallskip}-{\widetilde}\sigma_\mu& 0\cr}$$ or equivalently $$\label{Qdef}
Q = q^\mu \sigma_\mu = 