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

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.