Consider the Dirac equation (2) of §13 with $m=0$. Then equations (5) and (6) of §13 decouple, so it is enough to consider just one, say (5) of §13. This is the Weyl equation \begin{equation} \tilde{\PPP}\psi = 0 \label{Weyl} \end{equation} where we have written $\psi$ instead of $\theta$. In matrix notation, it is straightforward to show that the momentum $p^\mu$ of a solution of the Weyl equation must be null: ($\ref{Weyl}$) says that the $2\times2$ Hermitian matrix $\PPP$ has $0$ as one of its eigenvalues, which forces $\det(P)=0$, which is precisely the condition that $p^\mu$ be null.

Note that $\PPP$ is a complex matrix; it contains only one octonionic direction. But a $2\times2$ complex Hermitian matrix with determinant $0$ can be written as \begin{equation} \label{SolI} \PPP = \pm\theta\theta^\dagger \end{equation} where $\theta$ is also complex. The general solution of ($\ref{Weyl}$) is \begin{equation} \label{SolII} \psi = \theta\xi \end{equation} where $\xi\in\OO$ is arbitrary. It follows immediately from ($\ref{SolII}$) that \begin{equation} \label{Proportional} \psi\psi^\dagger = \pm |\xi|^2 \PPP \end{equation} which says that the vector constructed from $\psi$ is proportional to $\PPP$.

Since there are still just 2 octonions in all, $\xi$ and the components of $\theta$ (and hence also those of $\PPP$) belong to a quaternionic subalgebra of $\OO$. Thus, for solutions ($\ref{SolII}$), the Weyl equation ($\ref{Weyl}$) itself becomes quaternionic!

We can assume without loss of generality that this quaternionic subalgebra is the one containing $k$ and $\ell$. We therefore have \begin{equation} \PPP = p^t \II + p^x \SIGMA_x + p^y \SIGMA_y + p^z \SIGMA_z + p^k \SIGMA_k + p^{k\ell} \SIGMA_{k\ell} \end{equation} We can further assume, by a rotation in the plane containing $k$ and $k\ell$ if necessary, that $p^{k\ell}=0$. Writing $m=p^k$ brings $\PPP$ precisely to the form ($\ref{DiracWeyl}$)! The octonionic Weyl equation, describing massless spin-$1\over2$ particles in 10 dimensions, can therefore be reduced to the complex Dirac equation, describing, in general, massive spin-$1\over2$ particles in 4 dimensions. We will pursue this program in the next chapter.