Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W\_J = \langle s: s \in J\rangle \subseteq W$.


Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$.


**Proposition.** For each $w \in W$ there is a unique $u \in W^J$ and $v \in W\_J$ such that $w = uv$. 
Moreover, it holds for these elements that $\ell(w) = \ell(u) + \ell(v)$. Also, $u$ is the unique element of smallest length in the coset $wW\_J = \{wx : x \in W\_J\}$.


**Question:** Does $W^J\subseteq \langle t\in T: t\not\in W\_J\rangle$, where $T$ is the set of reflections?



About generator of minimal length coset representatives
In fact the minimal length coset representatives are exactly the elements $w$ such that for all reflections $t\in W\_J$ we have $\ell(wt) \gt \ell(w) $. So the answer to your question is yes. Specifically, suppose $w\in W^J$ has the reduced word $s\_1\cdots s\_k$. Define
$$t\_i=s\_ks\_{k-1}\cdots s\_i\cdots s\_{k-1}s\_k$$
Then none of the $t\_i$ are in $W\_J$ and
$$t\_kt\_{k-1}\cdots t\_2t\_1=w$$


