Abstract
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(eAeB) are Lie polynomials. Our proof is based on a recurrence formula for these components and a lemma that states that if under certain conditions a commutator of a non-commuting variable and a given polynomial is a Lie polynomial, then the given polynomial itself is a Lie polynomial.
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