The Baker-Hausdorff theorem states that for two given elements x and y in an associative algebra, the equation exey = ez has a solution z which lies in the Lie algebra generated by x and y. The Magnus continuous analog gives an exponential solution to a linear operator differential equation. Both theorems are valid globally for free Lie algebras of formal power series. For algebras that are not free, however, both theorems are locally but not globally valid. Some examples are given. Necessary and sufficient conditions for global validity are discussed. A superior representation in terms of a finite product of exponentials is also given.
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James Cheng‐Chung Wei (1963) studied this question.