  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 $EIII$.

Rings and Algebras