We describe a method for solving the Maurer–Cartan structure equation associated
with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to
integration. We show that the method naturally adapts to two other interesting
situations: local symplectic realizations of Poisson structures, in which case our
method sheds light on the role of the Poisson condition as an obstruction to
realization; and the Maurer–Cartan structure equation associated with a
Lie algebroid, in which case we obtain an explicit formula for a solution to
the equation which generalizes the well-known formula in the case of Lie
algebras.
