Remark 3

Let $T'$ be another maximal torus of G, B (resp. $B'$) a basis of the root system $R(G,T)$ (resp. $R(G',T')$) (Chap. VI, §1, no. 5, Def. 2). There exist elements $g\in G$ such that Int $g$ maps T onto $T'$ and B onto $B'$, and these elements form a unique coset modulo Int(T) (since T and $T'$ are conjugate, we can assume that $T = T'$, and it suffices to apply Chap. VI, §1, no. 5, Remark 4 and Prop. 9 of no. 4). It follows that the isomorphism from T to $T'$ induced by Int $g$ is independent of the choice of $g$; consequently the same is true of $D_*$(Int $g$) and $D^*$(Int $g$). Paraphrasing Chap. VIII, §5, no. 3, Remark 2, mutatis mutandis, we can now define the canonical maximal torus of G, the canonical covariant and contravariant root diagrams of $G,. .$..