# Subalgebras of the rank two semisimple Lie algebras

## TL;DR

This paper provides a comprehensive classification of subalgebras within rank 2 semisimple Lie algebras, including classical and exceptional types, using a unified scheme over complex numbers.

## Contribution

It completes the classification of subalgebras for all rank 2 semisimple Lie algebras, including the exceptional G2, with a uniform approach.

## Key findings

- Classification of subalgebras of classical rank 2 Lie algebras
- Completion of G2 subalgebra classification by Mayanskiy
- Unified scheme for low-dimensional Lie algebra classification

## Abstract

In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the subalgebras of the classical rank 2 semisimple Lie algebras. Finally, Mayanskiy finished the classification of the subalgebras of the remaining rank 2 semisimple Lie algebra, the exceptional Lie algebra $G_2$. We identify subalgebras of the classification in terms of a uniform classification scheme of Lie algebras of low dimension. The classification is up to inner automorphism, and the ground field is the complex numbers.