The formulas for the various formula_56's is "not" the Baker–Campbell–Hausdorff formula. Rather, the Baker–Campbell–Hausdorff formula is one of various expressions for formula_56's "in terms of repeated commutators of formula_4 and formula_5". The point is that it is far from obvious that it is possible to express each formula_56 in terms of commutators. (The reader is invited, for example, to verify by direct computation that formula_61 is expressible as a linear combination of the two nontrivial third-order commutators of formula_4 and formula_5, namely formula_64.) The general result that each formula_56 is expressible as a combination of commutators was shown in an elegant, recursive way by Eichler.