Let $S$ be a scheme and let $G\to S$ be a reductive group scheme. Then $G$ admits a maximal torus etale-locally, and any two maximal tori are conjugate etale-locally, by Theorem 3.2.6 and Corollary 3.2.7 in [B. Conrad, Reductive group schemes] http://math.stanford.edu/~conrad/papers/luminysga3.pdf.

As pointed out in the book, below Proposition 3.1.9 and at the beginning of Chapter 4, the existence of maximal tori can be achieved Zariski-locally (cf. [SGA3, XIV, 3.20]). My question is the following

Let $T$ and $T'$ be maximal tori in $G$ defined over $S$, are they conjugate Zariski-locally?

In my case, $T$ is split over $S$, and $S$ is affine with $Pic(S)=0$.

I would appreciate any remarks!

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    $\begingroup$ Take $S=\mathrm{Spec}(\mathbb{R})$ and $G=\mathrm{SL}_2$. Then the standard diagonal torus and $\mathrm{SO}_2$ are maximal tori, and they are not conjugate over $\mathbb{R}$. $\endgroup$ – Laurent Moret-Bailly Jan 12 at 19:09
  • $\begingroup$ @Laurent Moret-Bailly, thank you so much! It's simple and clear! $\endgroup$ – Evgeny Jan 12 at 19:39
  • $\begingroup$ What happens if one also assumes that $T$ and $T'$ are Zariski-locally isomorphic? $\endgroup$ – Uriya First Jan 13 at 8:06
  • $\begingroup$ If both tori are split, the answer seems to be "yes", for any two Borel subgroups are Zariski-locally conjugate, and maximal tori inside Borel subgroups are also conjugate (even globally if the base is affine). $\endgroup$ – Victor Petrov Jan 13 at 15:39

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