Remark 2

Say that a subgroup C of G is radical if there exists a maximal torus S of G and a subset P of $R(G,S)$ such that $C =\bigcap_{\alpha\in P}$ Ker$\alpha$. It follows from Prop. 13 and Lemma 2 of no. 5 that the map $H \rightarrow C(H)$ induces a bijection from the set of connected closed subgroups of maximal rank to the set of radical subgroups of G. The inverse bijection is the map $C \rightarrow Z(C)_0$.