Let $G/\mathbb{Q}$ be a connected reductive group, let $G^{\text{ad}}$ be the adjoint group, let $G^{\text{der}}$ be the derived group and let $\rho\colon G^{\text{sc}} \to G^{\text{der}}$ be the simply connected cover. Let $G^{\text{ad}}(\mathbb{R})^{0}$ be the identity component (in the real topology) of $G^{\text{ad}}(\mathbb{R})$, let $G(\mathbb{R})\_{+}$ be the inverse image of $G^{\text{ad}}(\mathbb{R})^{0}$ under the natural map $G(\mathbb{R}) \to G^{\text{ad}}(\mathbb{R})$ and let $G(\mathbb{Q})\_{+}$ be the intersection of $G(\mathbb{R})\_{+}$ with $G(\mathbb{Q}$). Note that $\rho(G^{\text{sc}}(\mathbb{R})) \subset G(\mathbb{R})\_{+}$ because $G^{\text{sc}}(\mathbb{R})$ is connected.


**Question**: Let $H/\mathbb{Q}$ be an inner form of $G$ and let the notation be as above, is there a `natural isomorphism' of abelian groups $H(\mathbb{Q})\_{+}/H^{\text{sc}}(\mathbb{Q}) \simeq G(\mathbb{Q})\_{+}/G^{\text{sc}}(\mathbb{Q})$?


When $G^{\text{sc}}=G^{\text{der}}$ this is true: Let $Z$ be the center of $G$, let $\nu:G \to D$ be the maximal abelian quotient of $G$, define $D(\mathbb{R})^{\dagger}:=\operatorname{Im}(Z(\mathbb{R}) \to D(\mathbb{R}))$ and let $D(\mathbb{Q})^{\dagger}=D(\mathbb{R})^{\dagger} \cap D(\mathbb{Q})$. Then Lemma 5.10 of <https://www.jmilne.org/math/xnotes/svi.pdf> shows that 
\begin{align}
\nu(G(\mathbb{Q})\_{+}) = D^{\dagger}(\mathbb{Q}).
\end{align}
Note that because $H$ is an inner form of $H$, we can identify its center with $Z$ and its maximal abelian quotient with $\mu:H \to D$. Applying the lemma again we see that
\begin{align}
\mu(H(\mathbb{Q})\_{+})=D^{\dagger}(\mathbb{Q})
\end{align}
and we are done. I have tried to follow a similar strategy in the general case, by which I mean comparing to $Z(\mathbb{Q})/Z^{\text{sc}}(\mathbb{Q})$, but I havent been able to get it to work.



Quotienting $G(\mathbb{Q})_{+}$ by $G^{\text{sc}}(\mathbb{Q})$ and inner forms
The answer is **Yes**. We denote $K(G)=G({\mathbb Q})\_+/\rho G^{\rm sc}({\mathbb Q})$.
We compute $K(G)$; see the corollary below.
It is clear from the corollary that $K(G)$ is canonically isomorphic to $K(H)$.


We will use Section 3 of [M. Borovoi, Abelian Galois cohomology of reductive groups. Memoirs of the AMS 132 (1998), No. 626](https://www.tau.ac.il/~borovoi/papers/galofile.pdf),
although all necessary results can be found in Deligne's paper
*Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques,*
Proc. Sympos. Pure Math. 33, Part 2, pp. 247–289.


We consider the *crossed module* $(G^{\rm sc}\to G)$ and the *hypercohomology*
$$H^0\_{\rm ab}({\mathbb Q},G):=H^0({\mathbb Q},G^{\rm sc}\to G),$$
where $G$ is in degree 0;
see the Memoir. By definition $H^0\_{\rm ab}({\mathbb Q},G)$ is a group.
We consider the abelian crossed module $(Z^{\rm sc}\to Z)$, where $Z=Z(G)$ and 
 $Z^{\rm sc}=Z(G^{\rm sc})$.
The morphism of crossed modules
$$(Z^{\rm sc}\to Z)\,\longrightarrow\,(G^{\rm sc}\to G)$$
is a *quasi-isomorphism*, and hence it induces a bijection on hypercohomology,
permitting us to identify $H^0\_{\rm ab}({\mathbb Q},G)$ with the abelian group $H^0({\mathbb Q},Z^{\rm sc}\to Z)$.
We conclude that $H^0\_{\rm ab}({\mathbb Q},G)$ is naturally an abelian group and that it does not change under inner twisting of $G$.


The short exact sequence
$$1\to(1\to G)\to (G^{\rm sc}\to G)\to (G^{\rm sc}\to 1)\to 1$$
(where $(G^{\rm sc}\to 1)$ is not a crossed module) induces a hypercohomology exact sequence
$$ G^{\rm sc}({\mathbb Q})\to G({\mathbb Q})\to H^0\_{\rm ab}({\mathbb Q},G)\to H^1({\mathbb Q},G^{\rm sc}),$$
where 
$${\rm ab}^0\colon G({\mathbb Q})\to H^0\_{\rm ab}({\mathbb Q},G)$$
 is the *abelianization map*.
This permits us to identify $G({\mathbb Q})/\rho G^{\rm sc}({\mathbb Q})$ with the kernel
$${\rm ker}[H^0\_{\rm ab}({\mathbb Q},G)\to H^1({\mathbb Q},G^{\rm sc})]$$
(yes, this kernel is a subgroup of the abelian group $H^0\_{\rm ab}({\mathbb Q},G)$ ).
This kernel might change under inner twisting of $G$, because $H^1({\mathbb Q},G^{\rm sc})$ changes under inner twisting.


By definition, 
$G({\mathbb R})\_+=Z({\mathbb R})\cdot\rho G^{\rm sc}({\mathbb R})$, and hence
$$G({\mathbb R})\_+/\rho G^{\rm sc}({\mathbb R})={\rm ab}^0(Z({\mathbb R}))\subset
{\rm ker}[ H^0\_{\rm ab}({\mathbb R},G)\to H^1({\mathbb R}, G^{\rm sc})].$$
We see that $K(G):=G({\mathbb Q})\_+/\rho G^{\rm sc}({\mathbb Q})$ can be identified with the preimage of ${\rm ab}^0(Z({\mathbb R}))\subset H^0\_{\rm ab}({\mathbb R},G)$ in ${\rm ker}[H^0\_{\rm ab}({\mathbb Q},G)\to H^1({\mathbb Q},G^{\rm sc})]$.



> 
> **Lemma.** The preimage of ${\rm ab}^0(Z({\mathbb R}))\subset H^0\_{\rm ab}({\mathbb R},G)$ in ${\rm ker}[H^0\_{\rm ab}({\mathbb Q},G)\to H^1({\mathbb Q},G^{\rm sc})]$
>  coincides with the preimage of ${\rm ab}^0(Z({\mathbb R}))$ in $H^0\_{\rm ab}({\mathbb Q},G)$.
> 
> 
> 


*Proof.*
Let $\xi\in H^0\_{\rm ab}({\mathbb Q},G)$ lie in the preimage of
$${\rm ab}^0(Z({\mathbb R}))\subset
{\rm ker}[ H^0\_{\rm ab}({\mathbb R},G) \to H^1({\mathbb R}, G^{\rm sc})].$$
Then the image of $\xi$ in $H^1({\mathbb R},G^{\rm sc})$ is trivial, and therefore,
the image of $\xi$ in $H^1({\mathbb Q},G^{\rm sc})$ lies in the kernel of the localization map
$$ H^1({\mathbb Q}, G^{\rm sc})\to H^1({\mathbb R},G^{\rm sc}).$$
By the Hasse principle for simply connected groups, this kernel is trivial.
Thus the image of $\xi$ in $H^1({\mathbb Q},G^{\rm sc})$ is trivial, and hence $\xi$ lies in the preimage of ${\rm ab}^0(Z({\mathbb R}))$ in ${\rm ker}[H^0\_{\rm ab}({\mathbb Q},G)\to H^1({\mathbb Q},G^{\rm sc})]$, as required.



> 
> **Corollary.** *The abelianization map ${\rm ab}^0\colon G({\mathbb Q})\to H^0\_{\rm ab}({\mathbb Q},G)$ with kernel $\rho G^{\rm sc}({\mathbb Q})$ induces a canonical isomorphism between the abelian groups $K(G):=G({\mathbb Q})\_+/\rho G^{\rm sc}({\mathbb Q})$
>  and the preimage of ${\rm ab}^0(Z({\mathbb R}))\subset H^0\_{\rm ab}({\mathbb R},G)$ in $H^0\_{\rm ab}({\mathbb Q},G)$.*
> 
> 
> 


