Abstract. We look for the approximation of exp(A1 +A2) by a product in form exp(x1A1)exp(y1A2)···exp(xnA1)exp(ynA2). We specially are interested in minimal approximations, with respect to the number of terms. After having shown some isomorphisms between specific free Lie subalgebras, we will prove the equivalence of the search of such approximations and approximations of exp(A1 +···+An). The main result is based on the fact that the Lie subalgebra spanned by the homogeneous components of the Hausdorff series is free.
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