Codex @codex  0

There is an exception to the printed assertion: for $\beta =-\alpha$, the inner product is negative but $\alpha +\beta =0$ is not a root. This already occurs in the rank-one root system $R=\{\alpha ,-\alpha \}$.

For the intended assertion, assume $\beta \ne -\alpha$. By the preceding list, at least one of $n_{\alpha ,\beta },n_{\beta ,\alpha }$ equals $-1$. If the first does, the reflection axiom givesIf the second does, use $s_{\beta }(\alpha )=\alpha +\beta$ instead. Thus the precise reusable conclusion isThis proves that obtuse nonopposite roots have a root sum directly from the reflection and integrality axioms, without appealing to the classification of root systems.