Theorem 2

a$) (X(T),X(T/(T\cap D(G)),R(G,T)))$ is a reduced root diagram; its Weyl group consists of the $X(w)$ for $w\in W$; the group X(C(G)) is isomorphic to the quotient of X(T) by the subgroup generated by $R(G,T)$.

b$) (\Gamma (T), \Gamma (C(G)_0),R^{\vee}(G,T))$ is a reduced root diagram; its Weyl group consists of the $\Gamma (w)$, for $w\in W$; the group $\pi_1(G)$ is isomorphic to the quotient of $\Gamma (T)$ by the subgroup generated by $R^{\vee}(G,T)$.

c) If each of the $\mathbf{Z}$-modules X(T) and $\Gamma (T)$ is identified with the dual of the other (no. 2, Prop. 3), each of the preceding root diagrams is identified with the inverse of the other.

Denote by $D^*(G,T)$ the diagram $(X(T),X(T/(T\cap D(G)),R(G,T)))$ and by $D_*(G,T)$ the diagram $(\Gamma (T), \Gamma (C(G)_0),R^{\vee}(G,T))$; these are called the contravariant diagram and the covariant diagram of G (relative to T), respectively.