Let $G$ be a connected, reductive group over a global field $k$. I am trying to understand why $G_v = G \times_k k_v$ is quasisplit for almost all places $v$ of $k$. There are several equivalent formulations of quasi-split, but none of them seem to be easy to work with:

1 . $G_v$ is quasisplit if and only if every/at least one minimal parabolic subgroup is solvable.

2 . Let $S_v$ be a maximal $k_v$-split torus of $G_v$, and let $T_v$ be a maximal torus of $G_v$ which is defined over $k_v$ containing $S_v$. There exist systems of positive roots for $T_v$ and $S_v$ such that positive roots of $T_v$ restrict to nonnegative characters. Let $\Delta$ be a base for a system of positive roots on $T_v$. Then $G_v$ is quasisplit if and only if none of the $\alpha \in \Delta$ are trivial on $S_v$.

3 . $G_v$ is quasisplit if and only if the following holds: whenever $P$ is a parabolic subgroup of $G_v \times_{k_v} \overline{k_v}$, $\gamma \in \textrm{Gal}((k_v)_s/k_v)$, and $\gamma(P)$ and $P$ are conjugate, then $P$ is conjugate to a parabolic subgroup which is defined over $k_v$.

The first two formulations seem especially difficult to work with. For example, in the second formulation, I see no way to control how the $S_v$ are chosen. For the third one, I thought one might be able to proceed by choosing in the beginning a Galois extension $E$ of $k$ over which $G$ splits, and for each finite place $v$ of $k$, choosing a place $w$ of $E$ lying over $v$. Then $G_v$ splits over $E_w$, and for the Galois action one can work with the subgroup

$$\textrm{Gal}(E_w/k_v) = \{ \sigma \in \textrm{Gal}(E/k) : \sigma \mathfrak P_w = \mathfrak P_w \}$$

where $\mathfrak P_w$ is the prime of $\mathcal O_E$ corresponding to $w$.

I would appreciate any hint or references. Thank you.

Reductive Group Schemesin the proceedings of the 2011 Luminy summer school on SGA3 see 3.1.9(1) and 3.1.12 for spreading-out over some $O_{k,S}$ and 5.2.11(3) for the existence and smoothness of the scheme of Borel subgroups). $\endgroup$ – nfdc23 Jun 5 '17 at 2:49