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:

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?

anymaximal torus in a quasi-split group is quasi-split in your sense (consider a compact torus in the split group $\mathrm{SL}(2,\mathbb{R})$ ). $\endgroup$ – Mikhail Borovoi Jun 6 '16 at 8:393more comments