I am currently reading the chapter 3- Root Systems of John Humphreys book on Lie Algebra. I am trying to recover my set of roots from knowledge of Cartan integers. (I am considering the base $\triangle$, root system $(\Phi,E)$.) \
The book says: We can start with the roots of height 1.(simple roots). \ To see this, We take any $\alpha_i\neq \alpha_j$, the integer $r$ for $\alpha_j$-string through $ \alpha_i$ is $0$ because $\alpha_i-\alpha_j$ is not a root and string is unbroken. So, $q=-<\alpha_i,\alpha_j>$.(This part is clear to me).\
I want to write down all roots of height 2 and further heights also. To find that, book says that we can determine all roots $\alpha$ of height 2 and hence all the integers $<\alpha,\alpha_j>$.This part is not clear to me. How they are connecting further root heights to integers $<\alpha,\alpha_j>$. \ And how to proceed for further height roots like for height 3,4 and so on by this...\ If someone can explain, it will be a great help!