Remark 1

Let $\mathfrak{c}$ be the centre of $\mathfrak{g}_{\mathbf{C}}$; then $\mathfrak{c}= L(C(G))_{(\mathbf{C})}$. We have the following relations between the diagrams of G relative to T and the direct and inverse root systems of the split reductive algebra $(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}}):$

a) The canonical isomorphism from $\mathbf{C}\otimes \Gamma (T)$ to $\mathfrak{t}_{\mathbf{C}}$ induces a bijection from $\mathbf{C}\otimes \Gamma (C(G)_0)$ to $\mathfrak{c}$ and a bijection from $1\otimes R^{\vee}(G,T)$ to $2\pi i.R^{\vee}(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$.

b) The canonical isomorphism from $\mathbf{C}\otimes X(T)$ to the dual $\mathfrak{t}^*_{\mathbf{C}}$ of $\mathfrak{t}_{\mathbf{C}}$ induces a bijection from $\mathbf{C}\otimes X(T/(T\cap D(G)))$ to the orthogonal complement of $\mathfrak{t}_{\mathbf{C}}\cap \mathscr{D}(\mathfrak{g})_{\mathbf{C}}$, and a bijection from $1\otimes R(G,T)$ to $R(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$.