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Recursively defined coefficients for the commutator-free form of the Campbell-Baker-Hausdorff formula are shown to apply to the commutator version of this formula, and a new relation on these coefficients is derived. Let x and y be noncommuting indeterminates over the rational numbers. The Campbell-Baker-Hausdorff formula asserts that exey ez, where z is a formal series in x and y with rational coefficients, and in fact that z is a Lie element, meaning that it may be written as a formal series in iterated commutators of x and y. (a) An attractive paper of Karl Goldberg published twenty-six years ago presents z as a commutator-free series in x and y:

Cyclic relations and the Goldberg coefficients in the Campbell-Baker-Hausdorff formula