Recover Lie algebra bracket from the root system

Question

I have coordinates of the $n$ roots in the $d$-dimensional space forming the root system of the $(n+d)$-dimensional Lie algebra. I want to implement the algorithm to recover the Lie bracket of the algebra.

1. The Lie bracket between the elements of the Cartan subalgebra is, by definition, zero: $$[x_i, x_j] = 0.$$
2. The Lie bracket between the $i$-th cartan subalgebra element and a root element is equal to the $i$-th coordinate of the root times the root element: $$[x_i, e_{\vec{\alpha}}] = \vec{\alpha}_i \cdot e_{\vec{\alpha}}. $$
3. How do I recover the Lie bracket between two roots? It can be shown to be proportional to the root element corresponding to the vector addition in the root space. I am interested in the proportionality coefficient.

Question: given two roots $\vec{\alpha}$ and $\vec{\beta}$ what is the easiest way to compute the proportionality coefficient $c$ in

$$ [e_{\vec{\alpha}},\, e_{\vec{\beta}}] = c \cdot e_{\vec{\alpha}+\vec{\beta}}. $$

Update: question 2: when $\vec{\beta} = - \vec{\alpha}$ the Lie bracket is proportional to the linear combination of the Cartan subalgebra elements, right? Is it only true if $\vec{\beta} = - \vec{\alpha}$? How to calculate the proportionality coefficients (are they just coordinates of the root vector)?

P.S. I know this involves solving the recursive relation, but I am just looking for a general "easy to use" formula or rule for $c$.

Answer 1

If I'm not mistaken, what you are looking for is naturally set in what is usually called a Chevalley basis and, in particular, Chevalley's structure constants $N_{\alpha, \beta}$. Up to their signs, they are determined relatively easily from the root system, as we have

$(N_{\alpha, \beta})^2 = (r+1)^2$

where $r$ is the maximal integer such that $\beta-r\alpha$ is a root. See most books on that topic or T. Tao's blog entry (lemma 42). Of course we can write this as

$N_{\alpha, \beta} = \pm(r+1)$

but determination of the signs for each pair needs more subtle considerations. There is an algorithm going back to Carter, see his book on Simple groups of Lie type. These notes of Casselman's give an overview about various approaches.