Let $G$ be a connected reductive group defined over a number field $K$ and $G^{der}$ its derived subgroup.

Let $\mathbb{A}_K$ denote the adeles of $K$.

Then for $G=GL_n$ we have $[GL_n(\mathbb{A}_K),GL_n(\mathbb{A}_K)]=SL_n(\mathbb{A}_K)=GL_n^{der}(\mathbb{A}_K)$. I'm interested in what generality this holds, in other words I'd like to ask:

**Question 1**: When is the commutator subgroup $[G(\mathbb{A}_K),G(\mathbb{A}_K)]$ equal to $G^{der}(\mathbb{A}_K)$?

As I think this question is really a local one, so let me put it this way:

Let $K_v$ be a local field of char 0. $G$ a reductive group over $K_v$, $G^{der}$ its derived subgroup.

**Question 2**: Is the commutator subgroup $[G(K_v),G(K_v)]$ equal to $G^{der}(K_v)$?

These questions came up when I wanted to understand 1-dimensional automorphic representations of unitary groups coming from a division algebra with an involution of the second kind and I realized I didn't know what the abelianizations of the adelic points of the groups in question were.