We propose a new approach to building log-canonical coordinate charts for any simply-connected simple Lie group $G$ and arbitrary Poisson-homogeneous bracket on $G$ associated with Belavin-Drinfeld data. Given a pair of representatives $r$, $r′$ from two arbitrary Belavin--Drinfeld classes, we build a rational map from $G$ with the Poisson structure defined by two appropriately selected representatives from the standard class to G equipped with the Poisson structure defined by the pair $r$, $r′$. In the $A_n$ case, we prove that this map is invertible whenever the pair $r$, $r′$ is drawn from aperiodic Belavin-Drinfeld data and apply this construction to recover the existence of a regular complete cluster structure compatible with the Poisson structure associated with the pair $r$, $r′$. This is  joint work with M. Shapiro and A. Vainshtein.