Can every positive root of a Coxeter group be written as a simple root and a positive root?

Question

Can every positive root of a Coxeter group be written as a simple root and a positive root? I think that this is possible. For example, in type $B_2$, the set of positive roots are $\alpha_1, \alpha_2, \alpha_1+\alpha_2, \alpha_1+2\alpha_2$. The positive root $\alpha_1+2\alpha_2$ can be written as $\alpha_2 + (\alpha_1+\alpha_2)$. Are there some reference about this result? Thank you very much.

Answer 1

This is not always possible for general Coxeter groups, where the root systems are not required to have integral coefficients. For example, for $I_2(5)$ there is an irrational number $z$ so that the positive roots are $\alpha_1$, $\alpha_2$, $\alpha_1+z\alpha_2$, $z\alpha_1+\alpha_2$, and $z\alpha_1+z\alpha_2$, with the final root being the counterexample.

However, in the integral case the simple roots are characterized as being the only positive roots $\alpha$ such that if $\beta$ is a positive root, then $\alpha-\beta$ is not a root. This can lead you to the result you want. In the non-integral case we must generalize this somewhat: $\alpha$ is simple if and only if there do not exist a positive root $\beta$ and a positive real number $a$ such that $\alpha-a\beta$ is a positive root.