= Solution
Use the <positive-root criterion for Coxeter length>: for a simple root $\alpha\in\Delta$,
$$
\ell(ws_\alpha)>\ell(w)
\quad\Longleftrightarrow\quad
w\alpha\in\Pi.
$$
The hypothesis $w\Delta\subseteq\Pi$ therefore says that every simple generator is a right ascent of $w$. If $w\ne1$, a <reduced expression in a Coxeter group> for $w$ has a final simple generator $s_\alpha$, and deleting it gives
$$
\ell(ws_\alpha)=\ell(w)-1,
$$
a contradiction. Hence $w=1$.
If $w$ stabilizes $\Delta$ setwise, then $w\Delta=\Delta\subseteq\Pi$, so the result just proved gives $w=1$. Thus the stabilizer of every fundamental system is trivial.
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