Roots of height 1 are necessarily simple.

Question

Suppose $\Phi$ is a root system, $\Pi \subset \Phi$ is a fundamental system (let $\Pi = \{r_1,...,r_l\}$). Now any root $r \in \Phi$ is a linear combination of the elements of $\Pi$ with all the coefficients either non-negative or non-positive: $$r = \sum\limits_{i=1}^{l} \lambda_i r_i.$$ Define $h : \Phi \longrightarrow \mathbb{Z}$ to be $$h(r) = \sum\limits_{i=1}^{l} \lambda_i.$$ It is well-known that $h(r)=1$ if and only if $r$ is a fundamental root. Okay, one direction is obvious: if $r$ is fundamental, then $h(r) = 1$. But why does it work in the opposite direction? I.e. I do not understand, why if $h(r)=1$, then $r$ is necessarily simple? Why does the situation like $r = \frac12 r_1 + \frac12 r_2$ cannot happen?

Answer 1

Because for any root $r \in \Phi$ can be written as linear combination of the elements of $\Pi$ with all the integer coefficients either non-negative or non-positive.