to the eigenvalues of ${\widehat}{L}_z$ which are not real, and in fact not in ${{\mathbb C}}$. We emphasize that the spin operators [*are*]{} self-adjoint (with respect to (\[SelfAdj\])). However, over the octonions it is not true that all the eigenvalues of Hermitian matrices, are real [@Eigen; @NonReal]; the case of self-adjoint operators is similar.

How does it affect the traditional interpretation of quantum mechanics to have simultaneous eigenstates of all 3 spin operators? The essential feature which permits this is that only one of the eigenvalues is real, and only real eigenvalues correspond to observables. Thus, from this point of view, the reason that the spin operators fail to commute is not that they do not admit simultaneous eigenstates, but rather that their [*eigenvalues*]{} fail to commute!

Furthermore, while (right) multiplication by a (complex) phase does not change any of the real eigenvalues, the nonreal eigenvalues do depend on the phase, since the phase doesn’t commute with the eigenvalue! Does this allow one [*in principle*]{} to determine the exact spin orientation, even if no corresponding measurement exists?

**DISCUSSION** {#DISCUSSION}
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We have shown how the massless Dirac equation in 10 dimensions reduces to the (massive and massless) Dirac equation in 