Sextonions, Zorn matrices, and $$\mathbf {e}_{\mathbf{7} \frac{\mathbf{1}}{\mathbf{2}}}$$ e 7 1 2

Abstract

By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field $\mathbb{C}$). This allows for an explicit construction, in terms of Jordan pairs, of the non-semisimple Lie algebra $\mathbf{e_{7 \frac12}}$, intermediate between $\mathbf{e_{7}}$ and $\mathbf{e_{8}}$, as well as of all Lie algebras occurring in the sextonionic row and column of the extended Freudenthal Magic Square.