2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/113880The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series H=ln(e^X e^Y) for non-commuting X,Y. Formally H lives in the graded completion of the free Lie algebra L generated by X,Y. We present a closed explicit formula for H=ln(e^X e^Y) in a linear basis of the graded completion of the free metabelian Lie algebra L/[[L,L],[L,L]].10 pages, the paper was completely reworked. The metabelian BCH formula is interpreted as a linear part of a deeper formula for ln(e^X e^Y)Quantum AlgebraAlgebraic Topology17B01text