Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.

From *"A note on generators for arithmetic subgroups of algebraic groups"* by Raghunathan, we have that for every ideal $I\lhd\mathbb Z$, $$\Gamma(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$$ is of finite index in $G(\mathbb Z)$.

>Is it also true for every ideal $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Specifically, I'm looking at $G=SL_n$  

Where:

 - $\Phi$ denote the root system of $G$ with respect to $T$;
 - $U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$;
 - $\alpha$ is an element that is integral over $\mathbb Z$;
 - if $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(\mathrm{mod}\,I)\}$ where $H(A):=H\cap \mathrm{GL}_n(A)$.