I quote Bjorner and Brenti, "Combinatorics of Coxeter Groups."
We begin with a simple geometric lemma. Let $m \geq 3$ be an integer, let $\gamma = \pi/m$, and let k, k' be real numbers such that k k' = 4 \cos^2 \gamma. Choose basis vectors $\beta$ and \beta' in the Euclidean plane such that the angle between $\beta$ and \beta' equals $\gamma$ and their lengths are related by
|\beta'| = \frac{2 \cos \gamma}{k} |\beta|
|\beta| = \frac{2 \cos \gamma}{k'} |\beta'|.
Let $r$ (resp $r'$) denote the orthogonal reflection in the line spanned by $\beta$ (resp $\beta'$).
Lemma
The coordinates (q, q') of a point q\beta + q'\beta' are transformed as follows by the orthogonal reflections:
r'(q, q') = (-q, q' + kq)
r(q, q') = (q + k' q', -q')
This seems crazy to me -- we define a variable $m$ which is never used again and place unnecessary restrictions on it; pull expressions for $k$ and k' out of nowhere, only to have them turn out to be exactly what we need later. Not only that, but we don't even find out why we're defining all these variables in this way until much farther down the page. If I were to write this out, I would do something like this:
Let $\beta$ and \beta' be linearly independent vectors in $\mathbb{R}^2$ which form an acute angle. Let $r$ (resp $r'$) denote orthogonal reflection about $\beta$ (resp $\beta'$). Then it is easy to see that
r \beta' + \beta' = (2|\beta'| \cos \gamma)\, \hat\beta,
and similarly
r' \beta + \beta = (2 |\beta| \cos (-\gamma))\, \hat\beta'.
Then we could define $k$ and k' and write out matrices for $r$ and r' in this basis. This isn't the only place I've seen this sort of presentation, either, so I'm guessing there must be some reason behind it that I'm missing. Can anyone explain why this is presented this way?