I have had a lot of questions lately about root systems and rationality questions. Several people have referred me to the article *Groupes Reductifs* by Borel and Tits, which I am slowly reading. In the meantime, I had a question about relative roots for which I hope there is a simple explanation.

$G$ is a connected reductive group defined over a field $F$, $T$ is a maximal torus of $G$, $A_0$ the maximal $F$-split subtorus of $T$, $X_0$ the kernel of the restriction map $X = X(T) \rightarrow X(A_0)$, and $\Phi \subseteq X$ the set of roots of $T$ in $G$. Let $\mathfrak g$ be the Lie algebra of $G$.

Let $\Phi_0 = \Phi \cap X_0$, and let $\overline{\Phi}$ be the image of $\Phi - \Phi_0$ in $X/X_0 \cong X(A_0)$. Then $\overline{\Phi}$ is called the set of restricted roots of $T$ in $G$.

On the other hand, I have seen the restricted roots defined to be $\Phi(A_0,G)$. Here $\alpha \in X(A_0)$ lies in $\Phi(A_0,G)$ if and only if there exists a nonzero $X \in \mathfrak g$ (or maybe it is required to be in $\mathfrak g(F)$, I'm not sure) such that $\textrm{Ad } t X = \alpha(t)X$ for all $t \in A_0$.

It is clear that $$\overline{\Phi} \subseteq \Phi(A_0,G)$$ On the other hand, the converse inclusion seems to be asserting the following: for each $\alpha \in \Phi(A_0,G)$, and any $\chi \in X$ whose restriction to $A_0$ is $\alpha$, there exists an element $\chi_0 \in X_0$ such that $\chi + \chi_0 \in \Phi$. Should this be the case?

Linear Algebraic Groups(Springer GTM 26, 1991) as well as the older Tits survey at the 1965 AMS Summer Institute in Boulder (AMS PSPM 9, 1967). Both sources include useful worked examples, some over local fields. What you ask about is seen to be true by following their definitions closely, though it's a complicated story and spills over into their follow-up IHES paper in 1972. $\endgroup$