List of simple roots in the H-basis for various Lie algebras? There are four usual bases one can use to express the roots and weights of a given algebra.


*

*The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$.

*The $\omega$-basis, where we write the roots and weights in terms of the fundamental weights $\omega_i$. The coefficients in this basis are often called Dynkin labels.

*The orthogonal-basis where one embeds the root/weight-space into a bigger Euclidean space. (See this question)

*The $H$-Basis where the coefficients for each weight or root correspond to the eigenvalues of the Cartan generators $H_i$


While I'm able to find list of the simple-roots in the $\alpha$-, the $\omega$- and orthogonal bases in almost any book, I'm struggeling for two days now to find a list of simple roots for groups like $A_4=SU(5)$ in the $H$-basis. 
Does a list of this kind exist somewhere? Any book, paper or lecture note suggestion would be awesome!
 A: The $H$-basis is based off the generalized Gell-Mann Matrices. In $N$ dimensions, there are $N-1$ diagonal matrices, called Cartan Generators. They are generalized as \begin{equation} [H_m]_{ij} = \frac{1}{\sqrt{2m(m+1)}}\left(\sum^m_{k=1}\delta_{ik}\delta_{jk}-m\delta_{i,m+1}\delta_{j,m+1} \right) .\end{equation}
Note that in this equation $i$ and $j$ are set based on your $N$ while $m=1,2,3, ... N-1$. The weights are the eigenvalues of these matrices with $N-1$ dimension. They can be generalized as well to \begin{equation} [w_m]^{j} = \frac{1}{\sqrt{2m(m+1)}}\left(\sum^m_{k=1}\delta_{jk}-m\delta_{j,m+1} \right) .\end{equation} Take notice that $[w^j]_m=[H_m]_{jj}$. Now finally the simple roots can be built:\begin{equation} \alpha^i=w^i-w^{i+1} \end{equation} for $i=1...N-1$
Edit based on OP's update:
I wrote some code for this a while ago and here are the 4 sources I used to learn it
[1] https://www.math.stonybrook.edu/~kirillov/mat552/liegroups.pdf
[2] https://arxiv.org/abs/1206.6379
[3] Georgi, Howard (1999) Lie Algebras in Particle Physics. Reading, Massachusetts: Perseus Books. ISBN 0-7382-0233-9.
[4] R. Slansky, Group theory for unified model building
