# How to find all the roots of a Lie Algebra starting from Cartan marix, following Georgi's book?

I am reading chapter 8 of Georgi's book, in particular sections 8.8-8.11, where he shows a diagrammatic procedure to find all the roots starting from simple roots. I understand that at each step, we find p by subtracting the entry in box from q, and if p is nonzero, we move to next level. I also understand that q for the simple roots is either 2 or 0. However, how do I calculate q for the next levels? For example, for eqn. (8.57), q=1 1. How is that calculated?

Let's take (2,-1). You know that $q=2, p=0$ for $\alpha_1$. For $\alpha_2$ you know that $q-p=-1$. Since $\alpha_1-\alpha_2$ is not a root, $q=0, p=1$. Let's then add $\alpha_2$, giving (1,1). We know that $p=0, q=1$ for $\alpha_2$ on this level. Starting from the other root (-1,2) you would find that $p=0, q=1$ for $\alpha_1$ acting on (1,1) so we are at the top of the spectrum. In general you can find all the information of the k+1th level by starting from the kth level.
• I must be missing something elementary here. Could you please explain how I get $p=0, q=1$ when I add $\alpha_1$ and $\alpha_2$? – textureguy Mar 18 '18 at 19:08