We first give a new direct geometric definition of a Lie bracket of two undirected curves on a surface which was found originally by Wolpert and Goldman in the eighties. and which is referred to here as the GW bracket. The geometric picture reveals readily certain nontrivial known results, unveils new ones and also motivates an unexpected but likely true conjecture. The conjecture reduces a hard to obtain geometric property, disjointness, to a structured algebraic calculation. The new result for the GW-bracket is that its center consists precisely of the obvious central elements, namely the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to the boundary components or punctures. We also prove in two short steps that there is no cancellation of geometric terms in the GW bracket of two curves if one curve is simple: firstly by showing that a cancelling pair of terms must have supplementary angles between geodesic representatives, by noting secondly, earthquaking along the simple geodesic changes these angles monotonically, this leading to the contradiction. Concerning the conjecture, we have substantial computer evidence suggesting that if the GW-bracket of two undirected curves is zero then these curves have disjoint representatives. This possibility was unanticipated by the history. The GW-bracket and an initial non-cancellation result go back to more algebraic work of first Wolpert on Poisson Lie algebras and then Goldman on his Lie algebras of directed and of undirected curves, all in the 80's.