Gradings on the simple real Lie algebras of types $G_2$ and $D_4$

TL;DR

This paper classifies all possible group gradings on the simple real Lie algebras of types G2 and D4, including fine gradings and gradings by a fixed group, over real closed fields.

Contribution

It provides a complete classification of group gradings on G2 and D4 Lie algebras over real closed fields, including fine and fixed-group gradings.

Findings

Classification of fine gradings up to equivalence

Classification of G-gradings up to isomorphism

Explicit descriptions of grading structures

Abstract

We classify group gradings on the simple Lie algebras of types $G_2$ and $D_4$ over the field of real numbers (or any real closed field): fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism.