On the commutator map for real semisimple Lie algebras

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 $p=2$), $\mathfrak u^*_{2m+1}(\mathbb H)$ ($m\ge1$) and $EIII$.