There is a theorem saying that if we take a Zariski-dense subgroup $\Gamma$ in a semisimple real Lie group $G$, for example $G = \operatorname{SL}_d(\mathbb{R})$, the set of so-called "loxodromic" elements in $\Gamma$ is still Zariski-dense in $G$. (In the case of $G = \operatorname{SL}_d(\mathbb{R})$, "loxodromic" means "diagonalizable with eigenvalues of distinct modulus").

In order to prove it, one actually proves the same statement for semigroups. See Proposition 5.11 and Remark 5.14 in Y. Benoist and J.F. Quint's book "Random walks on reductive groups".