Low Order Terms of the Campbell-Hausdorff Series and the Kashiwara-Vergne Conjecture

The Kashiwara-Vergne conjecture states that the Campbell-Hausdorff series of a Lie algebra can be written using a certain couple of functions.In this paper we consider universal solutions which apply to every real finitedimensional Lie algebra.We prove that a universal solution of the Kashiwara-Vergne conjecture verifying a natural symmetry condition is unique up to order one.In the Appendix by the second author, this result is used to show that the solutions of the Kashiwara-Vergne conjecture for a quadratic Lie algebra which exist in literature are not universal.