Exercise 8.11 of Humphreys' Introduction to Lie Algebras and Representation Theory asks to prove that if $\alpha, \beta \in \Phi$ and $(\alpha,\beta) > 0$, then $\alpha - \beta \in \Phi$ (presumably it's implicitly assumed that $\beta \neq \alpha$). Is the converse true? It's conceivable that we could have a string $\alpha - \beta, \alpha, \alpha + \beta, \alpha + 2\beta \in \Phi$, so then $\alpha(h_\beta) = 1 - 2 = -1 < 0$, but I don't have a concrete example of this.
2026-03-27 23:19:58.1774653598
Root system for Lie algebras: If $\alpha, \beta, \alpha - \beta \in \Phi$ are all roots, is $(\alpha, \beta) > 0$?
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The converse is not necessarily true. You'll find an example of your "conceivable" string in the root system of type $G_2$. (Not with the ones that are called $\alpha$ and $\beta$ in this image; rather, to use your terminology, use as $\alpha$ e.g. the upper left short root and as $\beta$ the one that is called $\alpha$ in this image).