I have a question about a proposition from Casselman's notes on representation theory. Let $\mathfrak a$ be a finite dimensional vector space, and $\Sigma$ a reduced root system in the dual $\mathfrak a^{\ast}$ with Weyl group $W$ and base $\Delta$. Via the inverse transpose, $W$ is also a group of automorphisms of $\mathfrak a$.
For $\theta \subseteq \Delta$, let $\mathfrak a_{\theta} = \{ x \in \mathfrak a : \langle x,\alpha \rangle = 0 \textrm{ for all } \alpha \in \theta \}$. In this vector space we have the hyperplanes $\mathfrak a_{\theta} \cap \mathfrak a_{\alpha} = \{ x \in \mathfrak a_{\theta} : \langle x ,\alpha \rangle = 0\}$ for $\alpha \in \Sigma^+ - [\theta]$, which partition $\mathfrak a_{\theta}$ into facets. The facets of maximal dimension are called chambers, the facets of next maximal dimension are called faces etc. (the usual notation from Bourbaki Lie Groups and Lie Algebras chapter V).
The set $C_{\theta} = \{x \in \mathfrak a_{\theta} : \langle x,\alpha \rangle > 0 \textrm{ for all } \alpha \in \Delta - \theta \}$ is a chamber in $\mathfrak a_{\theta}$. It is proved (Proposition 1.2.2) that every other chamber in $\mathfrak a_{\theta}$ is equal to $w C_{\theta'}$ for a unique $\theta' \subseteq \Delta$ and $w \in W$ with $w(\theta') = \theta$.
Let $W(\theta,\theta') = \{ w \in W : w(\theta') = \theta\}$ (confusing notation I know). For $w \in W(\theta,\theta')$, the height of $w$ (with respect to $\theta'$) is the minimal gallery length from the chamber $wC_{\theta'}$ to $C_{\theta}$. As usual, a gallery is a collection of chambers, each sharing a face with the next.
The proposition I want to understand is this:
I don't know where to start with this because I do not know how to begin relating height to length. The only useful result I can think of is that for an elementary conjugation, i.e. a Weyl group element of the form $w_0 = w_{l,\Omega}w_{l,\theta}$ for $\Omega = \theta \cup \{\alpha\}$, we have $\ell(w_0w) = \ell(w_0) + \ell(w)$ for all $w \in W_{\theta}$ (Lemma 1.1.2).