It is useful, perhaps, to split the process into two steps. In the first step, one constructs the free Lie algebra: this is what one gets, if one mods out by all commutators, without specifying what the values of the commutators are. The second step is to apply the specific commutation relations from formula_135 The first step is universal, and does not depend on the specific formula_135 It can also be precisely defined: the basis elements are given by Hall words, a special case of which are the Lyndon words; these are explicitly constructed to behave appropriately as commutators.