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We give a new purely algebraic proof of the Baker-Campbell-Hausdorff theorem,which states that the homogeneous components of the formal expansion of$\log(e^Ae^B)$ are Lie polynomials. Our proof is based on a recurrence formulafor these components and a lemma that states that if under certain conditions acommutator of a non-commuting variable and a given polynomial is a Liepolynomial, then the given polynomial itself is a Lie polynomial.

A relatively short self-contained proof of the Baker-Campbell-Hausdorff theorem