Obtaining Cartan Matrices from Lie Algebras

Question

I have been studying Lie Algebras from chapter 13 of Conformal Field Theory by Di Francesco et al., I understand that the entire structure of a Lie algebra is contained in its Cartan Matrix. What I do not understand is how one calculates it in the general case. Some examples are given on page 504-507 of the book, but they start with the Cartan matrix as the definition of the Lie algebra. How does one, for example, start with the Lie algebra sp(4) of the symplectic group SP(4), and find that the Cartan matrix is given as equation 13.102 of the book?