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?