Scholium

The classification of connected compact Lie groups up to isomorphism is thus reduced to that of reduced root diagrams. The connected compact semi-simple Lie groups correspond to the reduced root diagrams $(M,M_0,R)$ such that $M_0= 0$; giving such a diagram is equivalent to giving a reduced root system R in a vector space V over $\mathbf{Q}$ and a subgroup M of V such that $Q(R)\subset M\subset P(R)$.