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

TL;DR

This paper introduces a novel construction of the sextonion algebra using Zorn matrices and explicitly constructs the intermediate Lie algebra e_{7 1/2} via Jordan pairs, expanding the understanding of exceptional Lie algebras.

Contribution

It provides the first explicit construction of the sextonion algebra and the intermediate Lie algebra e_{7 1/2} using Zorn matrices and Jordan pairs.

Findings

Explicit construction of sextonions over a6

Construction of e_{7 1/2} Lie algebra between e_7 and e_8

Identification of Lie algebras in the sextonionic row and column

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.