Thank you for your comments, they were very helpful. I came across the construction via the quaternions earlier but was somehow thrown off the tracks by a a confusing remark.

I would like to add that the definition via the transformation of the volume carries another problem: While one can probably work out most problems (signs etc.) over $\mathbb{R}$ the determinant is defined for linear transformations of vector spaces over arbitrary fields and there is no obvious way to define a unit-cube and its volume in (e.g.) $\mathbb{F}_2^n$.

In all groups I have checked so far my claim holds but I still have no real idea on how to approach even a proof in a case-by-case fashion one diagram-type after the other...

@JimHumphreys: Thank you for your comments. I found the article by Brink's when I first thought about this problem but to be honest I found it fairly difficult to extract anything (be it positive or negative) regarding my claim. Regarding your second comment: I would actually be perfectly content if I could prove it for finite and affine Coxeter groups (I am looking at reductive groups over local fields so no "weird" Coxeter groups should occur there).

I just checked $D_4$ and $D_5$ and in both of them my assertion holds. Certainly my assertion would be wrong if I claimed to get the centralizer of $s_n$ in all of $W$ in the described way (or by adding $s_n$ to $Z$) as for example $I_2(4)$ or the $D_n$ groups show. But since I am only interested in elements which omit $s_n$ in their reduced expressions the situation seems to be at least a little bit more complicated.

Thanks for your comment. I will have a look at the paper (I actually already found that during my first google search but didn't take much from it, but I will take a closer look). Regarding your first two sentences: The Coxeter-group of type $I_2(4)$ also has non-trivial center but the assertion is true nonetheless so I will need to look at $D_n$ to see what happens there (e.g. in $D_4$ my claim seems to be true).