Relative Root Datum of an F-Reductive Group
Let \(G\) be a connected reductive \(F\)-group and let \(S\subseteq G\) be a maximal \(F\)-split \(F\)-torus. Write \[ {}_{F}X=X^*(S),\qquad {}_{F}X^\vee=X_*(S). \] The nonzero weights of \(S\) on \(\operatorname{Lie}(G)\) form the set of relative roots \({}_{F}R={}_{F}R(G,S)\subseteq {}_{F}X\). With the coroots defined by the rank-one relative subgroups, the quadruple \[ ({}_{F}X,{}_{F}R,{}_{F}X^\vee,{}_{F}R^\vee) \] is a root datum. Its isomorphism class is independent of the choice of maximal \(F\)-split torus.
The relative root system may be non-reduced: it can happen that both \(\alpha\) and \(2\alpha\) occur as relative roots.
Proof
Choose a maximal \(F\)-torus \(T\supseteq S\). The absolute root system \(R(G,T)\) restricts to characters of \(S\), and the nonzero restrictions are the relative roots. The centralizer \(Z_G(S)\) accounts exactly for the absolute roots that restrict trivially.
For each relative root \(\alpha\), the rank-one subgroup obtained from the centralizer of \((\ker\alpha)^\circ\) contains an element of \(N_G(S)(F)\) acting on \(X^*(S)\) by the orthogonal reflection attached to \(\alpha\). These reflections preserve the set of relative roots and the corresponding coroots. The root-datum axioms follow from the absolute root datum after projecting the absolute character space onto the relative character space and using these rank-one reflection representatives. Conjugacy of maximal \(F\)-split tori gives independence of \(S\).