and (\[Qdef\]), (\[Weyl\]) finally takes the form $$\label{StringII}
{\widetilde}{P}\psi = 0$$ which is the octonionic Weyl equation. In matrix notation, it is straightforward to show that the momentum $p^\mu$ of a solution of the Weyl equation must be null: (\[StringII\]) says that the $2\times2$ Hermitian matrix $P$ has $0$ as one of its eigenvalues, [^6] which forces $\det(P)=0$, which is $$\label{StringI}
{\widetilde}{P}P = 0$$ which in turn is precisely the condition that $p^\mu$ be null.

Equations (\[StringI\]) and (\[StringII\]) are algebraically the same as the octonionic versions of two of the superstring equations of motion, as discussed in [@FairlieI; @FairlieII; @Sudbery], and are also the octonionic superparticle equations [@Schray]. As implied by those references, (\[StringI\]) implies the existence of a 2-component spinor $\theta$ such that $$\label{SolI}
P = \pm\theta\theta^\dagger$$ where the sign corresponds to the time orientation of $P$, and the general solution of (\[StringII\]) is $$\label{SolII}
\psi = \theta\xi$$ where $\xi\in{{\mathbb O}}$ is arbitrary. The components of $\theta$ lie in the complex subalgebra of ${{\mathbb O}}$ determined by $P$, so that (the components of) $\theta$ and $\xi$ (and hence also $P$) belong to a quaternionic subalgebra of ${{\mathbb O}}$. Thus, for solutions (\[SolII\]), the Weyl equation (\[Weyl\]) itself becomes 