Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $H\_1,H\_2\subset G$ that are conjugated by an element $g\in G(\mathbb{R})$, there exists a number field $K$ of degree at most $d$, such that $\exists$ $\sigma\in G(K)$ with $\sigma H\_1 \sigma^{-1}=H\_2$?


The motivation of this question is from [A finiteness property for semi-simple algebraic groups](https://mathoverflow.net/questions/237556/a-finiteness-property-for-semi-simple-algebraic-groups). According to the answer of that question, we know that $G$ has only finitely many semi-simple $\mathbb{Q}$-subgroups modulo $G(\mathbb{R})$ conjugation, now what I want to know is that if two semi-simple $\mathbb{Q}$-subgroups are conjugate, can we always choose a conjugation element inside a "small" number field.


To sum up, I would like to know if this $d$ exists and the corresponding proof. Thanks very much!



About the conjugation of semi-simple subgroups
The answer is YES. It suffices to assume that $H\_1$ and $H\_2$ are conjugate over $\mathbb{C}$ or, what is the same, that they are conjugate over $\overline{\mathbb{Q}}$.



> 
> **Theorem 1.** Let $G$ be a connected semisimple linear algebraic group defined over a number field $k$. 
>  There exists a natural number $d=d(G\_{\bar{k}})$ with the following property:
>  let $H\_1$ and $H\_2$ be two connected semisimple $k$-subgroups of $G$ that are conjugate over ${\bar{k}}$,
>  then there exists a finite field extension $K=K(H\_1,H\_2)$ of $k$ of degree $[K:k]\le d$ such that $H\_1$ and $H\_2$ are conjugate over $K$.
> 
> 
> 


EDIT: This resut is valid over any perfect field, see the end of the answer.


*Proof of Theorem 1.* Choose an element $g\in G({\bar{k}})$ such that $g\cdot H\_1\cdot g^{-1} =H\_2$.
Let $s\in\mathrm{Gal}({\bar{k}}/k)$, then $^s\!g\cdot H\_1\cdot \,^s\!g^{-1}=H\_2$.
Set $c\_s=g^{-1}\cdot \,^s\!g$, then $c\_s\in N({\bar{k}})$, where $N$ is the normalizer of $H\_1$ in $G$, and $c=(c\_s)\in Z^1(k,N)$.
Let $[c]\in H^1(k,N)$ denote the cohomology class of $c$. 


If $[c]=1$, then there exists $n\in N({\bar{k}})$ such that $c\_s=n^{-1}\cdot \,^s\!n$ for all $s\in\mathrm{Gal}({\bar{k}}/k)$.
Set $g'=gn^{-1}$, then $g'\in G(k)$ and $g'\cdot H\_1\cdot (g')^{-1} =H\_2$. We see that if $[c]=1$, then $H\_1$ and $H\_2$ are conjugate over $k$.


Since there are only finitely many conjugacy classes of connected semisimple subgroups in $G\_{\bar{k}}$, see Friedrich Knop's answer to 
[this question](https://mathoverflow.net/questions/237556/),
 we see that Theorem 1 follows from the next Theorem 2.



> 
> **Theorem 2.** Let $N$ be a linear algebraic group (not necessarily connected or reductive) over a number field $k$. The there exists $d=d(N\_{\bar{k}})$
>  such that any cohomology class $\xi\in H^1(k,N)$ can be killed by a finite field extension $K=K(N,\xi)$ of degree $[K:k]\le d$. 
> 
> 
> 


*Proof of Theorem 2.* We argue by dévissage. If $N\_1$ is a normal subgroup of $N$, and $N\_2=N/N\_1$, then one sees from the corresponding cohomology exact sequence 
that it suffices to prove the theorem for $N\_1$ and $N\_2$.


Let $N^0$ denote the identity component of $N$, then $N/N^0$ is a finite group. For a finite group the theorem is clearly true 
(over any perfect field, not necessary a number field), so it suffices to prove the theorem for $N^0$.


We see that we may assume that $N$ is connected (and reductive). Let $R=Z(N)^0$ denote the identity component of the center of $N$, then $R$ is a torus.
A torus (over any perfect field) splits over a field extension of bounded degree, see YCor's comment, so we may assume that $R$ is split, and then $H^1(k,R)=1$ by Hilbert's Theorem 90.
 Therefore, it suffices to prove the theorem for the connected semisimple group $N/R$.


Now let $N$ be a connected semisimple group, $S$ the universal covering of $N$, then we have a short exact sequence
$$ 1\to C\to S\to N\to 1$$
where $S$ is a simply connected semisimple group and $C$ is a finite central subgroup of $S$. We obtain a cohomology exact sequence
$$ H^1(k,S)\to H^1(k,N)\to H^2(k,C).$$


Now we shall use the assumption that $k$ is a number field. We may assume that $k$ has no real embeddings. Then by the Hasse principle for simply connected groups 
we have $H^1(k,S)=1$. Therefore, Theorem 2 follows from the next Proposition 3.



> 
> **Proposition 3.** Let $C$ be a finite abelian $k$-group over a number field $k$. Then there exists $d=d(C\_{\bar{k}})$ such that any cohomology class $\eta\in H^2(k,C)$ 
>  can be killed by a finite field extension $K=K(C,\eta)$ of degree $[K:k]\le d$.
> 
> 
> 


*Proof of Proposition 3.* We may assume that $C$ is constant (i.e, that $C({\bar{k}})=C(k)$. Further, we may assume that $C$ is cyclic.
Further, we may assume that $C=\mu\_n$ for some natural $n$. Then
$H^2(K,\mu\_n)=\mathrm{Br}(k)\_n$ (the subgroup of elements of the Brauer group of $k$ killed by multiplication by $n$).


Let $b\in \mathrm{Br}(k)\_n$. We may assume that $k$ has no real embeddings. We know from the class field theory that $b$ defines local invariants $\mathrm{inv}\_v(b)\in\frac{1}{n}\mathbb{Z}/\mathbb{Z}$
for all finite places $v$ of $k$, and these local invariants are nonzero only for a finite set $\Xi$ of places. 
By weak approximation there exists a Galois extension $K/k$ of degree $n$ such that $K\_v:=K\otimes\_k k\_v$ is a field for $v\in\Xi$, where $k\_v$ is the completion of $k$ at $v$.
Then the extension $K\_v/k\_v$ kills $\mathrm{inv}\_v(b)$ for all $v\in\Xi$, hence the extension $K/k$ of degree $n$ kills $b$. 
This completes the proofs of Proposition 3, Theorem 2, and Theorem 1.


EDIT: As Vladimir Chernousov has explained me, theorems 1 and 2 are valid over any perfect field $k$, not necessarily a number field. It suffices to prove that if $N$ is a connected reductive $k$-group, then any cohomology class $\xi\in H^1(k,N)$ can be killed by a field extension of bounded degree. We may assume that $N$ is a split group. Then by Steinberg's theorem (Theorem 11.1 in "Regular elements of semisimple groups", republished in Serre's "Galois Cohomology"), $\xi$ comes from some torus $T\subset N$, and hence, can be killed by an extension of bounded degree.


