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](https://en.wikipedia.org/wiki/Coxeter_group).


I have some questions:


1. Is it true that every finite reflection group consists of some (not necessarily hyperplane) reflections and some rotations?
2. 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](https://en.wikipedia.org/wiki/Root_system), see section 1.14 of Jim Humphreys' book "Reflection groups and Coxeter groups". 



About reflections of reflection groups
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.


