Let W be a Coxeter group (not necessarily finite), and let Π and Φ be the corresponding root basis and root system. Suppose that x ∈ Φ + and a ∈ Π such that the coefficient of a in x is not zero. Prove that this coefficient must be greater than or equal to 1. You may assume that Π is linearly independent.
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In type $\mathsf{B}_2=\mathsf{C}_2$, you have a root $\beta_1+\frac{\sqrt{2}}{2}\beta_2$ where $\Pi={\beta_1,\beta_2}$ and the coefficient of $\beta_2$ is $<1$. I take the geometric representation of $W$. – Christoph Mark May 07 '18 at 12:01