Given a Lie algebra I understand all the stuff about the Cartan-Weyl basis, the roots and stuff, so for example, for A2, the roots are

with simple roots

and

. The scalar products are

and

using the definition of the scalar product in this basis from which one calculates the Cartan matrix and angles between roots blah blah blah...

I am reading

'Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists' By Jürgen Fuchs, Christoph Schweigert and I do not understand the Dynkin basis. It says the quadratic form matrix is given by

and

where

is the cartan matrix,

is the co-root

and

is the weight in the dual space with

. From this, they claim

and the Dynkin basis is

.

How does one calculate the matrix elements

? It seems a recursive definition of

because it is written in terms of the inner product but to define the inner product in this basis one needs to already know

Does my confusion make sense to anyone and could someone please clear this up for me?