Sextonions, Zorn Matrices, and $\mathbf{e_{7 \frac12}}$

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.
Comment: version 2 : minor refinements, Eq. (1.2) removed, and App. A added, in which we prove that - on R - the split sextonions do not contain the divisional quaternions. 14 pages, 6 figures. version 3 : Refs. added, various refinements; matches published version on Lett. Math. Phys