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Consider the following definition of a root system from Brian C. Hall, Lie Groups, Lie Algebras, and Representations, GTM 222.

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By 4. it follows that $\omega_\alpha(R)\subset R$. But later on, the author claims:

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I agree that $\alpha = \omega_\alpha\cdot (\omega_\alpha \cdot \alpha)$ but I can not follow why this implies the onto statement.

Sigur
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    I think there is a typo in the book. It should be $\beta\in R\implies \beta = w_\alpha\cdot(w_\alpha\cdot \beta)$. That will give the desired onto implication. – typesanitizer Sep 20 '17 at 20:14
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    @theindigamer Hopefully, it is a typo. I think that the question is closed now. Anyway, we can prove the onto part using your equality. Thanks. – Sigur Sep 21 '17 at 11:04

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