On the commutator map for 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 $\mathfrak{s}{u}_{p,q}$ ($p$ or $q$ >1), $\mathfrak{s}{o}_{p,p+2}$ ($p$ odd or $p=2$), ${\mathfrak{u}}_{2m+1}^{\ast }(\mathbb{H})$ ($m\ge 1$) and $EIII$.