Remark 2

Let G be a connected compact Lie group, $\widetilde{D}(G)$ the universal covering of the group D(G) and $f: \widetilde{D}(G)\rightarrow G$ the composite of the canonical morphisms from $\widetilde{D}(G)$ to D(G) and from D(G) to G. Then the map $T\rightarrow f^{-1}(T)$ is a bijection from the set of maximal tori of G to the set of maximal tori of $\widetilde{D}(G)$; the inverse bijection associates to a maximal torus $\widetilde{T}$ of $\widetilde{D}(G)$ the maximal torus $C(G)_0.f( \widetilde{T})$ of G.