  We reformulate the Poisson structure discovered by Fock and Rosly on moduli
spaces of flat connections over marked surfaces in the framework of Poisson
structures defined by Lie algebra actions and quasitriangular $r$-matrices, and
we show that it is an example of a mixed product Poisson structure associated
to pairs of Poisson actions, which were studied by J.-H. Lu and the author. The
Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied
by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories
between Poisson and quasi-Poisson spaces.
