Is there a good reference for different Cartan Weyl bases for Lie algebras?

Question

I'd just be interested if there is a source that lists some common CW bases for lie algebras of $SU(N),\,SO(N), Sp(N)$ etc. with roots and weights etc. (I know it's not hard to calculate them by hand, but it's neither fun nor particularly instructive)

Answer 1

This can be found in lecture notes on Lie algebras, e.g., in section $8$, i.e., in particular in $8.1.2$ on page $58$ of these lecture notes by Harold Steinacker in terms of root vectors $X_{\alpha},Y_{\alpha},H_{\alpha}$. Also Serre's Theorem gives the explicit Lie brackets for $L$ associated to the root system with a common basis:

Serre's theorem