quaternionic.

Furthermore, it follows immediately from (\[SolII\]) that $$\label{Proportional}
\psi\psi^\dagger = \pm |\xi|^2 P$$ Comparing this with (\[MSq\]), we see that the vector constructed from $\psi$ is proportional to $P$, or in more traditional language $${\overline}\Psi\gamma^\mu\Psi \sim p^\mu$$ which can be interpreted as the requirement that the Pauli-Lubanski spin vector be proportional to the momentum for a massless particle.

**DIMENSIONAL REDUCTION AND SPIN**
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Choosing a Preferred Complex Subalgebra {#CHOOSING}
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The description in the preceding section of 10-dimensional Minkowski space in terms of Hermitian octonionic matrices is a direct generalization of the usual description of ordinary (4-dimensional) Minkowski space in terms of complex Hermitian matrices. If we fix a complex subalgebra ${{\mathbb C}}\subset{{\mathbb O}}$, then we single out a 4-dimensional Minkowski subspace of 10-dimensional Minkowski space. The projection of a 10-dimensional null vector onto this subspace is a causal 4-dimensional vector, which is null if and only if the original vector was already contained in the subspace, and timelike otherwise. The time orientation of the projected vector is the same as that of the original, and the induced mass is given by the norm of the remaining 6 components. Furthermore, the ordinary Lorentz group $SO(3,1)$ clearly sits inside the 