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We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras(that is, Lie algebras admitting an invariant scalar product) reduces to theproblem of representing the Campbell-Hausdorff series in the formln(e^xe^y)=x+y+[x,a(x,y)]+[y,b(x,y)], where a(x,y) and b(x,y) are Lie series inx and y. This observation explains the existence of explicit rational solutionsof the quadratic KV problem (see M. Vergne, C.R.A.S. 329 (1999), no. 9,767--772 and A. Alekseev, E. Meinrenken, C.R.A.S. 335 (2002), no. 9, 723--728arXiv:math/0209346), whereas constructing an explicit rational solution of thefull KV problem would probably require the knowledge of a rational Drinfeldassociator. It also gives, in the case of quadratic Lie algebras, a directproof of the Duflo theorem (implied by the KV problem).

On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras