We give a new purely algebraic proof of the Baker–Campbell–Hausdorff theorem, which states that the homogeneous components of the formal expansion of $log\left({\mathrm{e}}^A{\mathrm{e}}^B\right)$ 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.

中文翻译：

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Baker-Campbell-Hausdorff定理的相对较短的独立证明

我们给出了Baker-Campbell-Hausdorff定理的一个新的纯代数证明，该定理指出： $日志（{\mathrm{Ë}}^{\mathrm{一种}}{\mathrm{Ë}}^{乙}）$是李多项式。我们的证明基于这些分量的递推公式和一个引理，引理指出，如果在某些条件下非交换变量和给定多项式的交换子是Lie多项式，那么给定多项式本身就是Lie多项式。