I'm trying to prove the following result.

If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is not solvable, and $[\Gamma, \Gamma]$ is infinite.

So far, I've managed to prove that $\Gamma$ can't be abelian, since $\Gamma$ projects densely onto the maximal compact factor $\{e\}$, and a version of the Borel density theorem tells us that $\mathcal{C}_G(\Gamma) = Z(G)$, and $Z(G)$ is finite, but $\Gamma$ isn't, which rules out the possibility that $\Gamma$ is abelian. My question is whether I can refine this argument to prove that $\Gamma$ isn't solvable as well, or do I need some additional machinery to prove that? Thanks!