Proposition 13

Let H be a connected closed subgroup of G of maximal rank, and C its centre. Then C contains the centre of G, and H is the identity component of the centralizer of C.

Let S be a maximal torus of H. Since the centre of G is contained in S, it is contained in C. Put $L = Z(C)_0$; this is a connected closed subgroup of G containing H, hence is of maximal rank, and its centre is equal to C. Denote by $R_H$ and $R_L$ the root systems of H and L, respectively, relative to S; then $R_H\subset R_L\subset R(G,S)$. Since C(H) = C(L), Prop. 8 (no. 4) implies the equality $Q(R_H) = Q(R_L)$; but $Q(R_H)\cap R_L= R_H$ (Chap. VI, §1, no. 7, Prop. 23), so $R_H= R_L$ and H = L (Prop. 12).