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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!

Vats Y
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    Consider $A_3$ for example. $\langle \alpha_1,\alpha_2\rangle=-1$ implying that $\alpha=\alpha_1+\alpha_2$ is a root. Then calculate $\langle\alpha,\alpha_3\rangle$, and show that $\alpha_1+\alpha_2+\alpha_3$ is also a root. While you are at it, show that $\alpha_2+\alpha_3$ is a root, but no other positive roots are necessary (and don't exist in this root system). – Jyrki Lahtonen Jul 11 '21 at 09:39
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    Most of the connected Dynkin diagrams lead to somewhat boring root systems. But $E_6$, $E_7$, $E_8$ and $F_4$ are more fun :-) – Jyrki Lahtonen Jul 11 '21 at 09:40
  • Thanks for the same. I got some idea. Can you also explain why other roots in case of $A_3$ are not possible! Is it because if we talk about $\alpha_1+\alpha_3$, $<\alpha_1,\alpha_3>=0$ for example.? – Vats Y Jul 11 '21 at 10:33
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    Yeah. $\alpha_1$ and $\alpha_3$ are orthogonal. So Pythagoras says that the vector $\alpha_1+\alpha_3$ is longer than either of $\alpha_1,\alpha_3$. But in this root system all the roots have the same length. – Jyrki Lahtonen Jul 11 '21 at 13:44
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    Related: https://math.stackexchange.com/q/3265516/96384, https://math.stackexchange.com/q/1467330/96384. – Torsten Schoeneberg Jul 13 '21 at 20:02

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