About reflections of reflection groups For any finite crystallographic reflection group $W = \langle s_1, \ldots , s_n\rangle$,  every hyperplane reflection is of the form $ws_iw^{-1}$ for some $i$ and some $w \in W$. 
A finite crystallographic reflection group $W$ is a Coxeter group with the presentation
\begin{align}\label{Coxeter system}
S=\langle s_1, s_2, \ldots, s_n \mid (s_is_j)^{m_{ij}} = 1 \rangle,
\end{align}
where $(s_i)_{1 \leq i \leq n}$ is the set of simple reflections and $m_{ij} \in \{2,3,4,6\}$. The pair $(W,S)$ is called a Coxeter system.
I have some questions:


*

*Is it true that every finite reflection group consists of some (not necessarily hyperplane) reflections and some rotations? 

*What are the reduced words of reflections under a Coxeter system $(W,S)$?


For any finite reflection group, the number of hyperplane reflections is the number of positive roots in the corresponding root system, see section 1.14 of Jim Humphreys' book "Reflection groups and Coxeter groups". 
 A: 1) No. $W$ consists of elements of determinant 1 and -1. According to your wikpedia, all elements of determinant 1 are "rotations". Elements of determinant -1 are not necessarily reflections because they are not necessarily of order 2. Just think of a 4-cycle $(1,2,3,4)\in S_4$: his order is 4, not 2. It is a proper roto-reflection...
BTW, the terminology is confusing: a rotation can be a reflection! Think of a reflection across 2-codimensional subspace.
2) It is explained by Sam for (hyperplane) reflections. They are all of the form $ws_iw^{-1}$ for some $w\in W$, and they will have a reduced word of this kind.
Higher-dimensional (fixing a subspace of higher codimension) reflections can be figured out as well. They are just elements of order 2. I do not know their reduced words off the top of my head.
A: Every coxeter group consists of reflections.  An even number of reflections around two elements is a rotation, which is what $(s_is_j)^{m_{ij}}$ means.  The infinite coxeter groups consists of sets of parallel mirrors, which equate to a translation rather than a reflection.  
One might note the root lattice consists of perpendiculars to each of the mirrors, radiating from a point.  This produces a lattice.
The 'reduced word' is the shortest path between two points.  So if $ABABAB=1$ as in the hexagon, then $ABAB=BA$, where $BA$ is the reduced form.
