Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s_1\dots s_n$ with $s_i$ simple reflections, then $s_i(\gamma)=\gamma$ for $i=1\dots n$?
I can't see a reason why this shouldn't be true, but am unable to prove it.