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A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial

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Adjacent parabolic subgroups and proportionality to $\alpha^{\vee}$

Probably you have already know the answer. For those who are interested in it, a proof is given by Lemme 3.3.1 of La formule des traces tordue d’après le Friday Morning Seminar by Labesse and Waldspur …
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