Controllability of control systems on complex simple lie groups and the topology of flag manifolds

Let S be a subsemigroup with nonempty interior of a connected complex simple Lie group G. It is proved that S = G if S contains a subgroup G(α) ≈ Sl (2, $ \mathbb{C} $ ) generated by the exp $ {{\mathfrak{g}}_{{\pm \alpha }}} $ , where $ {{\mathfrak{g}}_{\alpha }} $ is the root space of the root α. 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(α) are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.