The compact exceptional Lie algebra $\mathfrak g^c_2$ as a twisted ring group

TL;DR

This paper introduces a new, highly symmetrical model of the compact Lie algebra c_2 as a twisted ring group, providing an orthogonal basis with integer structure constants and extending to the split Lie algebra c_2^*.

Contribution

It presents a novel twisted ring group model of c_2 that is self-contained and does not require prior knowledge of roots or octonions, with applications to both compact and split Lie algebras.

Findings

Provides an orthogonal basis with integer structure constants.

Generalizes Pauli and Gell-Mann matrices.

Models both c_2 and c_2^* as twisted ring groups.

Abstract

A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in $\mathfrak{su}(2)$ and of the Gell-Mann matrices in $\mathfrak{su}(3)$. As a bonus, the split Lie algebra $\mathfrak{g}^*_2$ is also seen as a twisted ring group.