For a Weyl group $W$, I would like to know whether each $w\in W$ can be expressed as $w=s\_{\alpha\_1}s\_{\alpha\_2}\cdots s\_{\alpha\_k}$ for some distinct positive roots $\{\alpha\_1, \alpha\_2, \cdots, \alpha\_k\}\subseteq \Phi^+$.


I know for type A, the above is true. 


Since $W(A\_n)\cong S\_{n+1}$ with the map $s\_{e\_i-e\_j}\mapsto (i \ j)$. For $w\in W(A\_n)$, let $\sigma\in S\_{n+1}$ be the image of $w$ under the map. 
By the fact that every permutation can be decomposed as a product of disjoint cycles in the form of $(a\_1\cdots a\_l)$. 
 And also the fact that $(a\_1\cdots a\_l)=(a\_1 \ a\_l)\cdots (a\_1 \ a\_2)$. We get $\sigma$ is a product of distinct transpositions. And distinct transpositions correspond to distinct reflections of the form $s\_{e\_i-e\_j}$. Therefore, the claim follows. 


I would like to know whether the above statement is also true for other types?



Each $w\in W$ can be expressed as product of distinct reflections?
The answer is "yes" and there is a geometric explanation.


Let $\mathcal{H}$ denote the set of hyperplanes corresponding to the reflections $s\_\alpha$ with $\alpha\in\Phi^+$ (note that $s\_\alpha=s\_{-\alpha}$), and let $\Sigma$ denote the connected components of $V\setminus\bigcup\_{H\in\mathcal{H}}H$ (where $V$ is the vector space where $\Phi$ lives). The elements of $\Sigma$ are the chambers of the Coxeter complex of $(V,\Phi)$, and it is well-known that $W$ acts simply transitively on $\Sigma$.


Fix some $C\_0\in\Sigma$, let $w\in W$, and put $C\_1=w(C\_0)$.
Then you can "walk" from the chamber $C\_0$ to $C\_1$ by crossing through the walls of the chambers (here, a wall means a codimension-$1$ face of a chamber). Every time you cross from $C$ into an adjacent chamber $C'$, you walk though some hyperplane $H\in \mathcal{H}$, and in that case you have $C'=s\_\alpha C$, where $\alpha\in \Phi^+$ corresponds to $H$.
Since you can walk from $C\_0$ to $C\_1=wC\_0$ without crossing the same hyperplane $H$ twice (Edit: e.g. take a walk minimizing the number of chambers visited, or go along a straight line in general position w.r.t. $\mathcal{H}$), it follows that $C\_1=s\_{\alpha\_t}\cdots s\_{\alpha\_1}C\_0$, where $\alpha\_t,\dots,\alpha\_1\in\Phi^+$ are distinct and correspond to the hyperplanes you crossed on your way from $C\_0$ to $C\_1$.
Thus, $w=s\_{\alpha\_t}\cdots s\_{\alpha\_1}$ with distinct $\alpha\_t,\dots,\alpha\_1\in \Phi^+$.


