If you have a finite, real reflection group, why can the length function $\ell(-)$ be interpreted as the codimension of the fixed subspace, or alternatively, as the number of eigenvalues different from $1$?
Everywhere leads me back to Carter's original 1972 paper, where it is unproven.
(promoting my comment to an answer to get this thread wrapped up).
This version of Carter's 1972 paper seems to have a proof (Lemma 2).
The key ingredients are the fact that the intersection of $k$ codimension one subspaces has codimension $\le k$ and the fact (Lemma 1) that an element of the Weyl group fixing (pointwise) a given set of vectors can be written as a product of reflections fixing the same set of vectors.