as a circle. The octonionic units can be grouped into (the imaginary parts of) quaternionic subalgebras in 7 different ways, corresponding to the 7 lines in the figure; these will be referred to as quaternionic triples. Within each triple, the arrows give the orientation, so that e.g. $$ij = k = -ji$$ Any three imaginary basis units which do not lie in a such a triple anti-associate. Note that any two octonions automatically lie in (at least one) quaternionic triple, so that expressions containing only two independent imaginary octonionic directions do associate. [*Octonionic conjugation*]{} is given by reversing the sign of the imaginary basis units, and the norm is just $$|p| = \sqrt{p {\overline}{p}}$$ which satisfies the defining property of a normed division algebra, namely $$|pq| = |p| |q|$$

We follow [@SUDBERY; @Chung] in representing real $(9+1)$-dimensional Minkowski space in terms of $2\times2$ Hermitian octonionic matrices. [^4] In analogy with the complex case, a vector field $q^\mu$ with $\mu=0,...,9$ can be thought of under this representation as a matrix $$Q = \pmatrix{~~q^+& {\overline}{q}\cr \noalign{\smallskip} q& ~~q^-}$$ where $q^\pm=q^0\pm q^9 \in{{\mathbb R}}$ are the components of $q^\mu$ in 2 null directions and $q=q^1 +q^2 i + ... + q^8 \ell \in{{\mathbb 