\right)$$ We therefore consider the eigenvalue problem $$\label{HEigen}
{\widehat}{L}_z \psi = \psi \lambda$$ with $\lambda\in{{\mathbb R}}$. It is straightforward to show that the real eigenvalues are $$\lambda_\pm = \pm {1\over2}$$ as expected. However, the form of the eigenvectors is a bit more surprising: $$\psi_+ = \pmatrix{A\cr kD\cr} \qquad\qquad
  \psi_- = \pmatrix{kB\cr C\cr}$$ where $A,B,C,D\in{{\mathbb C}}$ are any elements of the preferred complex subalgebra, and $k$ is any imaginary octonionic unit orthogonal to $\ell$, so that $k$ and $\ell$ anticommute. Thus, the components of spin eigenstates are contained in the quaternionic subalgebra ${{\mathbb H}}\subset{{\mathbb O}}$ which is generated by $\ell$ and $k$.

Therefore, if we wish to consider spin eigenstates, $\ell$ must be in the quaternionic subalgebra ${{\mathbb H}}$ defined by the solution. We can further assume without loss of generality that ${{\mathbb H}}$ takes the form given in (\[Cayley\]). Thus, the only possible nonzero components of $p_\mu$ are $p_t=p_0$, $p_x=p_1$, $p_k=p_4$, $p_{k\ell}=p_5$, $p_y=p_8$, and $p_z=p_9$, corresponding to the gamma matrices with components in ${{\mathbb H}}$. We can further assume (via a rotation in the ($k,k\ell$)-plane if necessary) that $p_5=0$, so that $$\label{PComplex}
P = \pi(P) + m \, \sigma^k$$ where $$\pi(P) = p_\alpha \sigma^\alpha \equiv {{\bf p}}$$ with $\alpha = 