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

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