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Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient algorithm to compute the absolute root system of $G$ with respect to $T$?

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    $\begingroup$ Are you assuming that $G$ itself is split, or at least quasisplit? If not (or even if so), then how is $G$ given to you? Then also, how is $T$ given to you—say, by specifying a cocycle in the absolute Weyl group? And I assume that you want not just the absolute root system of $G$, which doesn't depend on the torus, but also the Galois action on it? $\endgroup$
    – LSpice
    Commented Aug 27 at 1:14

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