Quotienting formula_137 by formula_155 has the effect of setting all Lie commutators defined in formula_137 to zero. One can see this by observing that the commutator of a pair of elements whose products lie in formula_157 actually gives an element in formula_155. This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank. The essence of the Poincaré–Birkhoff–Witt theorem is that it is always possible to do this, and that the result is unique.