Proposition 16

For every reduced root diagram D, there exists a connected compact Lie group G such that $D^*(G)$ (resp. $D_*(G)$) is isomorphic to D.

a) By replacing D, if necessary, by its inverse diagram, we are reduced to constructing G such that $D^*(G)$ is isomorphic to D. Put $D = (M,M_0,R)$; then $\mathbf{Q}\otimes M$ is the direct sum of $\mathbf{Q}\otimes M_0$ and the vector subspace V(R) generated by R. Moreover, since the inverse roots take integer values on M, the projection of M on V(R) parallel to $\mathbf{Q}\otimes M_0$ is contained in the group of weights P(R) of R, so that M is a subgroup of $M_0\oplus P(R)$ of finite index. Denote by $D'$ the diagram $(M_0\oplus P(R),M_0,R)$.

b) Let $\mathfrak{a}$ be a complex semi-simple Lie algebra whose canonical root system is isomorphic to $R\subset \mathbf{C}\otimes V(R)$ (Chap. VIII, §4, no. 3), and let $\mathfrak{g}_1$ be a compact real form of $\mathfrak{a}($§3, no. 2, Th. 1). Let $G_1$ be a simply-connected real Lie group whose Lie algebra is isomorphic to $\mathfrak{g}_1$; then $G_1$ is compact (§1, no. 4, Th. 1). Let $T_1$ be a maximal torus of $G_1$. By Th. 1, the diagram $D^*(G_1,T_1)$ is isomorphic to $(P(R),0,R)$.

c) Let $T_0$ be a torus of dimension equal to the rank of $M_0$; then $D^*(T_0,T_0)$ is isomorphic to $(M_0,M_0,\emptyset )$, so $D^*(G_1\times T_0,T_1\times T_0)$ is isomorphic to $D'$ (Example 2).

d) Finally, let N be the finite subgroup of $T_1\times T_0$ orthogonal to M. Put $G = (G_1\times T_0)/N, T = (T_1\times T_0)/N$. Then G is a connected compact Lie group, T a maximal torus of G, and $D(G,T)$ is isomorphic to D (Example 3).