Definition of the Dynkin Diagram (in Humphreys)

Question

I'm reading paragraph 11 in Humphreys' 'Introduction to Lie Algebras and Representation Theory'. The author defines Coxeter graphs and Dynkin diagrams for any rank-many distinct positive roots. He then claims that the Cartan matrix can be recovered from the Dynkin diagram which i do understand in case the distinct positive roots are simple. How can we do that in the general case? Is there a connection between the Dynkin diagram of a base and the diagram of 'random' roots?

Answer 1

The Cartan matrix and the Dynkin diagram deal only with simple roots. Knowing the relations between the simple roots is enough to determine the entire root system up to isomorphism. The purpose of the Dynkin diagram and the Cartan matrix are as shorthand for describing the root system and corresponding algebraic structures.