Let $\Delta$ be a Zariski dense finitely generated subgroup of $SL(3,\mathbb{R})$. Assume that $\Delta$ contains no element of finite order. Then, does there exist a finite-order element $A \in SL(3,\mathbb{R})$ such that the group generated by all the elements of $\Delta$ and $A$ contains $\Delta$ as a finite index subgroup? We assume that $A$ is not an identity.
Is there some known conditions for $\Delta$ that the above statement holds?