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. 



abelianization of adelic points of an algebraic group
I think question 2 has a positive answer when $G^{der}$ is simply connected [EDIT: and without anisotropic factor], but not in general. If $G=PGL\_d$ (so $G=G^{der}$) then $G(K)/[G(K),G(K)]=M/M^d$ where $M=K^\*$, this is not a trivial group in general.


Relevant refences should be the preliminary chapters in book by Margulis "discrete subgroups of semisimple Lie groups", as well as Platonov-Rapinchuk's book.


