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 $K\subset \mathbb{R}$ of degree at most $d$, such that $\exists$ $\sigma\in G(K)$ with $\sigma H\_1 \sigma^{-1}=H\_2$?


Note that we have already known that $K$ can be chosen as a number field of uniformly bounded degree, see: [About the conjugation of semi-simple subgroups](https://mathoverflow.net/questions/239022/about-the-conjugation-of-semi-simple-subgroups). Now I would like to figure out if $K$ can be further taken as a real field. Thanks very much!



A more precise description of conjugation of semi-simple subgroups
The answer is YES.



> 
> **Theorem 1.** Let $G$ be a connected semisimple linear algebraic group defined over a number field $k\subset{\mathbb{R}}$.
> 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 ${\mathbb{R}}$,
> then there exists a finite real field extension $K=K(H\_1,H\_2)\subset {\mathbb{R}}$ of $k$ of degree $[K:k]\le d$ such that $H\_1$ 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/),
Theorem 1 follows from the next Theorem 2 (as in [my previous answer](https://mathoverflow.net/questions/239022/about-the-conjugation-of-semi-simple-subgroups)).



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


Note that the analog of Theorem 2 for abelian varieties is [false](https://mathoverflow.net/questions/239396/algebraic-points-of-uniformly-bounded-degree-on-an-algebraic-variety).


We need three lemmas.



> 
> **Lemma 1.** Let $N$ be a *finite* $k$-group. Then any ${\mathbb{R}}$-point of $N$ is defined over a real number field of bounded degree.
> 
> 
> 


This is obvious.



> 
> **Lemma 2.** If $N$ is a *connected* linear algebraic $k$-group, then $N(k)$ is dense in $N({\mathbb{R}})$.
> 
> 
> 


See Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80, Cor. 3.5(iii).



> 
> *Lemma 3.* If $N$ is a *connected or abelian* linear algebraic $k$-group, then the localization map
> $$ H^1(k,N)\to H^1({\mathbb{R}},N) $$
> is surjective.
> 
> 
> 


See Sansuc's paper cited above, Lemma 1.6 and Lemma 1.12.


Note that the analog of Lemma 3 for nonabelian finite $k$-groups is an [open question](https://mathoverflow.net/questions/50570/real-approximation-for-homogeneous-spaces-of-linear-algebraic-groups).


With the help of Lemmas 1, 2, and 3, we use dévissage in order to reduce Theorem 2 to the following four propositions.



> 
> **Proposition 1.** Theorem 2 is true when $N$ is a finite $k$-group.
> 
> 
> 


Indeed, then any $k$-torsor $\mathcal{T}$ of $N$ is a finite $k$-scheme, hence any ${\mathbb{R}}$-point of $\mathcal{T}$ is real algebraic of bounded degree.


Now let $N$ be connected, then it splits over an algebraic extension $L\subset \mathbb{C}$ of bounded degree. Set $K=L\cap{\mathbb{R}}$.
After passing to $k=K$, we may assume that $N$ splits over $k$ or over a quadratic extension $L$ of $k$.
After passing to a real quadratic extension, we may assume that $k$ has only one real embedding.



> 
> **Proposition 2.** Theorem 2 is true when $N$ is a simply connected semisimple $k$-group.
> 
> 
> 


Indeed, under our assumptions the localization map
$$ H^1(k,N)\to H^1({\mathbb{R}},N) $$
is bijective (the Hasse principle for simply connected groups), hence $\xi=1$.



> 
> **Proposition 3.** Theorem 2 is true when $N$ is a $k$-torus.
> 
> 
> 


*Proof.* We assume that $N$ splits over a quadratic extension $L$ of $k$.
Then we may assume that $N$ is a nonsplit one-dimensional $k$-torus.
Write $L=k(\sqrt{-\alpha})$, where $\alpha\in k$, $\alpha>0$.
Then $N$ is given by the equation
$$ x^2+\alpha y^2=1,$$
and a $k$-torsor $\mathcal{T}$ of $N$ is given by the equation
$$ x^2+\alpha y^2=c,$$
where $c\in k^\times$.
By assumption $\mathcal{T}$ has an ${\mathbb{R}}$-point $(x\_1,y\_1)$,
and we may assume that $x\_1,y\_1>0$.
Choose $y\_2\in k$ such that $0< y\_2<y\_1$ and set $\zeta=\sqrt{c-\alpha y\_2^2}$,
then $K:=k(\zeta)$ is a real quadratic extension of $k$, and $\mathcal{T}$ has a $K$-point $(\zeta, y\_2)$. This $\mathcal{T}$ splits over a real quadratic extension, as required.



> 
> **Proposition 4.** Theorem 2 is true for $H^2(k,A)$, where $A$ is a finite abelian $k$-group.
> 
> 
> 


*Proof.* Set $A'=\mathrm{Hom}(A,\mathbb{G}\_m)$, the dual group. We say that $A$ is *split* over $k$ if the Galois group $G\_k$ acts trivially on $A'$.
As above, we may assume that $A$ splits over a quadratic extension $L$ of $k$ and that $k$ has only one real embedding.
Then, since $A$ splits over a quadratic extension, the Hasse principle for $H^1(k,A)$ and $H^2(k,A)$ holds.
The Galois group $G\_k$ acts on $A'$ via its quotient group $\Gamma=G\_{L/k}$ of order 2. We may assume that $A'$ is a *simple* $\Gamma$-module, i.e., is has no proper
$\Gamma$-submodules. Then $\#A=\ell$ is a prime number, and either $\Gamma$ acts on $A'$ trivially,
or $\ell$ is odd and $\Gamma$ acts on $A'$ by multiplication by $-1$.
The first case was treated in [my previous answer](https://mathoverflow.net/questions/239022/about-the-conjugation-of-semi-simple-subgroups): we can kill a cohomology class
$\eta\in H^2(k,A)$ that vanishes over $L$ by a real extension of degree $\ell$. Similarly, in the second case case one can kill a cohomology class $\eta$ that vanishes over $L$ by a real extension of degree $\ell$.
(One should use the Tate duality and the Tate-Poitou exact sequence, see Serre's book "Galois cohomology".)


This completes the proofs of Theorem 1 and Theorem 2.


