On the Poisson bracket on the free Lie algebra in two generators

We prove a combinatorial formula for the Poisson bracket of two elements of the free Lie algebra on two generators, which has a particularly nice cocycle form when the. two elements are Lie monomials containing only one y. By relating this cocycle form with the period polynomials introduced by Eichler-Shimura and Zagier, we completely describe and classify a set of fundamental relations in Ihara's stable derivation algebra, generalizing the first few cases of these relations which he had observed and computed by hand.