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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. 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