I'm going through the calculation of the constant term of Eisenstein series in Moeglin and Waldsburger's Spectral Decomposition and Eisenstein series book (section II.1.7), and am confused on a small detail. Let $P = MN$ and $P' = M'N'$ be standard parabolic subgroups of a connected, reductive group $G$ over a field $k$, identified with their $k$-points. If $w$ is an element of the Weyl group, for which $w.\alpha > 0$ for all simple roots of $M$, and $w^{-1}.\alpha > 0$ for all simple roots of $M'$, I'm trying to understand how to decompose
$$P' \cap wPw^{-1} \backslash P'$$ The calculation of the constant term appears to rely on the fact that a set of right coset representatives of $P' \cap wPw^{-1}$ in $P'$ is $m'n'$, where $m'$ runs through a set of right coset representatives of $P' \cap wMw^{-1} \backslash M'$, and $n'$ runs through a set of right coset representatives of $P' \cap m'^{-1}wNw^{-1}m' \backslash N'$. I have been trying for awhile to prove that this is true, but have gotten stuck. I would appreciate any hint, proof, or reference.
eisenstein-series, which had no other questions under its heading $\endgroup$