# The subalgebras of the rank two symplectic Lie algebra

## TL;DR

This paper completes the classification of all subalgebras of the rank 2 symplectic Lie algebra, including semisimple, Levi decomposable, and solvable subalgebras, providing a comprehensive understanding of its subalgebra structure.

## Contribution

It provides the first complete classification of solvable subalgebras of rak{sp}(4,c), completing the classification of all subalgebras of this algebra.

## Key findings

- Classified all solvable subalgebras of rak{sp}(4,c)
- Completed the classification of subalgebras of rank 2 semisimple Lie algebras
- Unified the understanding of subalgebra structures in rak{sp}(4,c)

## Abstract

The semisimple subalgebras of the rank $2$ symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{C})$ are well-known, and we recently classified its Levi decomposable subalgebras. In this article, we classify the solvable subalgebras of $\mathfrak{sp}(4,\mathbb{C})$, up to inner automorphism. This completes the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$. More broadly speaking, in completing the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$ we have completed the classification of the subalgebras of the rank $2$ semisimple Lie algebras.