Lemma 79.15.13. In Situation 79.15.4 assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.
Proof. The proof of this lemma is literally the same as the proof of Lemma 79.15.11 except that “strong splitting” needs to be replaced by “quasi-splitting” (2 times) and that the reference to Lemma 79.15.8 needs to be replaced by a reference to Lemma 79.15.10. $\square$
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