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We propose a new approach to building log-canonical coordinate charts for anysimply-connected simple Lie group $\G$ and arbitrary Poisson-homogeneousbracket on $\G$ associated with Belavin--Drinfeld data. Given a pair ofrepresentatives $r, r'$ from two arbitrary Belavin--Drinfeld classes, we builda rational map from $\G$ with the Poisson structure defined by twoappropriately selected representatives from the standard class to $\G$ equippedwith the Poisson structure defined by the pair $r, r'$. In the $A_n$ case, weprove that this map is invertible whenever the pair $r, r'$ is drawn fromaperiodic Belavin--Drinfeld data, as defined in~\cite{GSVple}. We further applythis construction to recover the existence of a regular complete clusterstructure compatible with the Poisson structure associated with the pair $r,r'$ in the aperiodic case.

A unified approach to exotic cluster structures on simple Lie groups