Consider the following question about forms of a given group that are embedded in a fixed group.
Fix for simplicity $k$ a perfect field, and $H\subsetneq G$ a pair of connected reductive $k$-groups, with $G$ being $k$-simple. It is possible that one finds other $k$-groups $L\subset G$ that are $k$-forms of $G$.
(1) How to classify those $k$-forms $L$ embedded in $G$ such that $L_\bar{k}$ is conjugate to $H_{\bar{k}}$ by $G(\bar{k})$?
(2) Given a $k$-form $L$ of $H$, can one always find a larger $k$-group $G$ containing both $H$ and $H'$, such that (1) holds for the triple $(H,L,G)$? Here $G$ is required to not normalize any $k$-factor of $H$.
To algebraize the formulation, take $L$ be a $k$-form of $H$ embedded also in $G$, and set the following $k$-groups: $N=N(H,G)$ the normalizer, and $T=T_G(H,L)$ the transporter from $H$ to $L$ in $G$. According to Demazure-Gabriel, Groupes algebriques, II, sect.1, 3.6, 3.7, $T$ is representable in this case and $T$ has a right action by $N$. Evaluating at $\bar{k}$ one sees that $T$ is a $N$-torsor for the etale topology, which. $T$ admits a $K$-rational point, $K$ being some finite extension of $k$, if and only if the $N$-torsor splits over $K$.
But the above arguments rely on the existence of a $k$-form $L$ embedded in $G$. A priori I don't know if such $k$-forms exists, because given a $\bar{k}$-subgroup of $G_\bar{k}$ conjugate to $H_\bar{k}$, I don't know if there is a criterion to assure that it is also defined over $k$. Such criteria provided, the question can be reduced to
(i) finding $k$-splitting of the torsor $T$;
(ii) classifying the torsors $T$ that come from $k$-forms embedded in $G$.
although (ii) is more or less repeating (1).
Many thanks indeed.
