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 $\gamma_2 = \begin{pmatrix} 1 & 0 \newline
 1 & 1 
 \end{pmatrix}$.