  In this note the notion of Poisson brackets in Kontsevich's "Lie World" is
developed. These brackets can be thought of as "universally" defined classical
Poisson structures, namely formal expressions only involving the structure maps
of a quadratic Lie algebra. We prove a uniqueness statement about these Poisson
brackets with a given moment map. As an application we get formulae for the
linearization of the quasi-Poisson structure of the moduli space of flat
connections on a punctured sphere, and thereby identify their symplectic leaves
with the reduction of coadjoint orbits. Equivalently, we get linearizations for
the Goldman double Poisson bracket, our definition of Poisson brackets
coincides with that of Van Den Bergh in this case. This can furthermore be
interpreted as giving a monoidal equivalence between Hamiltonian quasi-Poisson
spaces and Hamiltonian spaces.
