Structure constants from Dynkin diagram

Question

Is there an easy way to obtain the structure constants of a Lie Algebra starting from its Dynkin diagram?

Answer 1

Yes, there is. You can write down the Cartan matrix from the Dynkin diagram as usual, and then proceed with the Serre relations. See also here:

Computing information about a Lie algebra from cartan matrix

On the other hand, if you know the name of the root system, then you also know the matrix Lie algebra. In the above case it is $B_2$ and hence the matrix Lie algebra $\mathfrak{so}(5)$. This gives you the explicit Lie brackets by $[A,B]=AB-BA$.