Example 1

If G is semi-simple of rank 1, then $D^*(G,T)$ and $D_*(G,T)$ are necessarily isomorphic to one of the two diagrams $\Delta_2= (\mathbf{Z},0,\{2,-2\})$, $\Delta_1= (\mathbf{Z},0,\{1,-1\})$. If G is isomorphic to $\mathbf{S}\mathbf{U}(2,\mathbf{C}), D_*(G,T)$ is isomorphic to $\Delta_1$ (since G is simply-connected) so $D^*(G,T)$ is isomorphic to $\Delta_2$. If G is isomorphic to $\mathbf{S}\mathbf{O}(3,\mathbf{R}), D^*(G,T)$ is isomorphic to $\Delta_1$ (since $C(G) =\{1\}$), so $D_*(G,T)$ is isomorphic to $\Delta_2$.