Maximal reductions in the Baker–Hausdorff formula

A preliminary expression for the Baker–Hausdorff formula is found up to ninth order, i.e., a series expansion of z in terms of multiple commutators, where ez=exey with x and y noncommuting, up to ninth degree in x,y. By means of complete sets of linear relations between multiple commutators maximal reduction of the number of different multiple commutators in the series are obtained.