Let $\alpha$ be an integral element and set $k=\mathbb Q(\alpha)$, and $U$ is a $k$-defined unipotent subgroup of $\operatorname{GL}_n(k)$ with $k-rank\geq 2$.

Let $Λ$ be a subgroup of $U(\mathcal{O}_k)$ which is Zariski-dense in $U$, I'm trying to prove there is an ideal $I\lhd\mathcal{O}_k$ such that $U(I)\subset Λ$.

The congruence subgroup property applies for unipotent groups under the $\mathcal{O}_S$ ring so I think my question comes down to:

Is $Λ$ of finite index in $U(\mathcal{O}_k)$?

Specifically, I'm looking at $U$ to be all the upper triangular matrices with ones on the diagonal in $\mathrm{SL}_n(k)$ for $n\geq 3$