On the commutator map for real semisimple Lie algebras

TL;DR

This paper establishes new conditions under which the commutator map in real semisimple Lie algebras is surjective, and applies these to identify which simple algebras have this property.

Contribution

It provides novel sufficient conditions for surjectivity of the commutator map in real semisimple Lie algebras and determines the algebras for which it holds.

Findings

Surjectivity of the commutator map for most simple algebras

Identification of exceptions like  su_{p,q} and  so_{p,p+2}

New criteria for the surjectivity in real semisimple Lie algebras

Abstract

We find new sufficient conditions for the commutator map of a real semisimple Lie algebra to be surjective. As an application we prove the surjectivity of the commutator map for all simple algebras except $\mathfrak su_{p,q}$ ($p$ or $q$ >1), $\mathfrak so_{p,p+2}$ ($p$ odd or $p=2$), $\mathfrak u^*_{2m+1}(\mathbb H)$ ($m\ge 1$) and $EIII$.