Defining amalgams of compact Lie groups

For n ≥ 2 let ∆ be a Dynkin diagram of rank n and let I = {1, . . . , n} be the set of labels of ∆. A group G admits a weak Phan system of type ∆ over C if G is generated by subgroups Ui, i ∈ I , which are central quotients of simply connected compact semisimple Lie groups of rank one, and contains subgroups Ui,j = 〈Ui, Uj〉, i 6= j ∈ I , which are central quotients of simply connected compact semisimple Lie groups of rank two such that Ui and Uj are rank one subgroups of Ui,j corresponding to a choice of a maximal torus and a fundamental system of roots for Ui,j . It is shown in this article that G then is a central quotient of the simply connected compact semisimple Lie group whose complexification is the simply connected complex semisimple Lie group of type ∆.