Write a positive nonsimple root as
$$
\alpha=\sum_{\beta\in\Delta}n_\beta\beta,
\qquad n_\beta\geq0.
$$
If $(\alpha,\beta)\leq0$ for every simple root with $n_\beta>0$, then
$$
(\alpha,\alpha)=\sum_\beta n_\beta(\alpha,\beta)\leq0,
$$
which is impossible. Hence $(\alpha,\beta)>0$ for some simple $\beta$. The root-string property then gives $\alpha-\beta\in\Phi$, and its simple-root coefficients remain nonnegative. This is the <simple-root subtraction lemma>.

Induct on the height $\sum n_\beta$. Applying the induction hypothesis to $\alpha-\beta$ and appending $\beta$ writes
$$
\alpha=\alpha_1+\cdots+\alpha_k
$$
so that every partial sum is a root.

Finally let $\alpha$ be simple and let $\gamma\ne\alpha$ be positive. In the simple-root expansion of
$$
s_\alpha(\gamma)=\gamma-
\langle\gamma,\alpha^\vee\rangle\alpha,
$$
all coefficients except possibly that of $\alpha$ are unchanged, and at least one of those unchanged coefficients is positive. Since a root has coefficients all of one sign, the image cannot be negative. Thus $s_\alpha$ permutes $\Phi^+\setminus\{\alpha\}$, as stated by <action of a simple reflection on positive roots>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 3 / a / ii / Solution

If a positive root $\alpha$ is not simple, then $\alpha-\beta$ is a positive root for some simple root $\beta$. Iteration writes every positive root as a sum of simple roots whose successive partial sums are roots.


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 Incoming links: Simple-root subtraction lemma