Let $G$ be a semisimple algebraic group, and let $\Gamma$ be a finitely generated subgroup. Given the generators of $\Gamma$, is there a good way to determine if $\Gamma$ is Zariski-dense in $G$?

For example, let $\Gamma \subseteq \mathrm{PSL}_2$ where $ \displaystyle \Gamma = \left\langle \gamma_1 , \gamma_2 \right\rangle$ and $\gamma_1 = \left( \begin{array}{ll} 1 & i, \\ 0 & 1 \end{array} \right)$ and $\gamma_2 = \left( \begin{array}{ll} 1 & 0, \\ 1 & 1 \end{array} \right)$.

(These should be 2 by 2 matrices, but the code doesn't seem to be working.)

Let $G$ be a semisimple algebraic group, and let $\Gamma$ be a finitely generated subgroup. Given the generators of $\Gamma$, is there a good way to determine if $\Gamma$ is Zariski-dense in $G$?

For example, let $\Gamma \subseteq \mathrm{PSL}_2$ where $ \displaystyle \Gamma = \left\langle \gamma_1 , \gamma_2 \right\rangle$ and $\gamma_1 = \begin{pmatrix} 1 & i \newline 0 & 1 \end{pmatrix}$ and $\gamma_2 = \begin{pmatrix} 1 & 0 \newline 1 & 1 \end{pmatrix}$.