Maximal tori determining the algebraic groups. (Q2496431)

Let \(k\) be a finite field, a global field, or a local non-Archimedean field. Let \(H_1\) and \(H_2\) be split, connected, semisimple algebraic groups over \(k\). It is proven that if \(H_1\) and \(H_2\) share the same set of maximal \(k\)-tori, up to \(k\)-isomorphism, then their Weyl groups are isomorphic, hence \(H_1\) and \(H_2\) are isogenous except for a switch of a certain number of factors of type \(B_n\) and \(C_n\). A recent result of Gille implies a similar result for other fields \(k\).