Let $k$ be a perfect field.


Recall that an algebraic torus $T$ over $k$ is called *quasi-split* if there exists some finite étale $k$-algebra $A$ such that
$$T \cong \mathrm{R}\_{A/k} \mathbb{G}\_m.$$


A reductive group $G$ over $k$ is called *quasi-split* if it contains a Borel subgroup $B$ over $k$.


A priori, I see no reason why these two definitions should be related; Indeed any algebraic torus is quasi-split in the second sense (take $B=T$), but generally not in the first sense. However, I have seen it claimed, or at least implicitly used, in proofs that for *semisimple* $G$, the latter implies the former. Namely:



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> > Let $G$ be a semisimple algebraic group over $k$ with a maximal torus $T$. If $G$ is quasi-split, then is $T$ quasi-split? Does the converse hold?
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Quasi-split tori and algebraic groups
A torus $T$ is quasi-split if its character group is a permutation representation for the Galois group. So a counterexample to your question is: let $G$ be the quasi-split group $SO(n+1,n-1)$, $n\ge2$, $k=\mathbb R$. The weight lattice is $\mathbb Z^n$ with Galois action
$$
(x\_1,\ldots,x\_{n-1},x\_n)\mapsto(x\_1,\ldots,x\_{n-1},-x\_n)
$$
This is not a permutation module.


On the other hand, the Galois action on $T\subseteq B\subseteq G$ of a quasi-split group is induced by an action on the simple roots. These form a basis if $G$ is of adjoint type. For simply connected groups one can play the same game with fundamental representations. So:


If $G$ is quasi-split, and either simply connected or of adjoint type then the maximal $k$-torus sitting in a $k$-Borel is quasi-split.


I have the suspicion that the two notions of "quasi-splitness" developed independently.


