Let $B$ be a Borel subgroup of a connected reductive algebraic group $G$ with maximal torus $T$. Let $\Phi = \Phi(G,T)$ be the root system, $\Delta$ be the basis of $\Phi$ defined by $B$, and $W$ be the Weyl group. Let $w \in W$. Then $\dim(BwB) = \ell(w) + \dim(B)$ with $\ell$ being the length function.
Now let $I_k \subseteq \Delta$ ($k = 1,2$), and let $P_k = M_kN_k$ be the corresponding standard parabolic subgroups. Here, $N_k$ is the unipotent radical, and $M_k$ is a Levi factor (chosen such that $T \subset M_k$). Let $W_1,W_2 \subset W$ be the corresponding Weyl groups.
My question: Is there a way to write down a formula for $d := \dim(M_1wP_2)$, or even to obtain $$d = \ell(w_{max}) - \ell(w_{min}) + \dim(B)?$$ Here, $w_{min}$ and $w_{max}$ are the unique elements of minimal and maximal length, respectively, in the double coset $W_1wW_2$.