  Let $S$ be subsemigroup with nonempty interior of a complex simple Lie group
$G$. It is proved that $S=G$ if $S$ contains a subgroup $G(\alpha) \approx
\mathrm{Sl}(2,\mathbb{C}) $ generated by the $\exp \mathfrak{g}_{\pm \alpha}$,
where $\mathfrak{g}%_{\alpha}$ is the root space of the root $\alpha $. The
proof uses the fact, proved before, that the invariant control set of $S$ is
contractible in some flag manifold if $S$ is proper, and exploits the fact that
several orbits of $G(\alpha)$ are 2-spheres not null homotopic. The result is
applied to revisit a controllability theorem and get some improvements.
