In the classical setting, we can define automorphic forms on $\text{SL}_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups.

I have encountered the lifting of automorphic forms to the adeles (so to automorphic representations) for $\Gamma$ being a congruence subgroup or, more generally, the unit group of an order in a quaternion algebra. I am wondering what are the precise conditions that make this lift possible in general.

Can we "lift" any classical arithmetic subgroup to a compact open subgroup over the adeles? I understand how this might happen when the subgroup is associated to an order in an algebra (the matrix algebra or a division algebra). But there are other constructions of lattices $\Gamma$, especially in higher rank. What I am maybe asking is whether simply arithmeticity of the subgroup is enough to make full use of adelic lifts (if they even exist).