The center of the enveloping algebra and the Campbell-Hausdorff formula

Let g be a Lie algebra. In this Note, we define g-valued functions F(x, y) and G(x, y) on g + g, such that x + y - log(e(x)e(y)) = (e(ad) (x) - 1)F(x, y) + (1 - e(-ad) (y))G(x, y). Furthermore, if g is a quadratic Lie algebra, we prove an identity for the trace of the matrix (ad x) partial derivative(x)F + (ad y)partial derivative(y)G. This identity was conjectured in [4] for any Lie algebra g, and proved when a is a soh,able Lie algebra. This result implies (see [4]) that Duflo's isomorphism [2] extends naturally to convolution algebras of invariant distributions on the group G and the Lie algebra a. (C) 1999 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.