In the case that formula_1 arises as the Lie algebra of a real Lie group, one can use left-invariant differential operators to give an analytic proof of the Poincaré–Birkhoff–Witt theorem. Specifically, the algebra formula_185 of left-invariant differential operators is generated by elements (the left-invariant vector fields) that satisfy the commutation relations of formula_1. Thus, by the universal property of the enveloping algebra, formula_185 is a quotient of formula_68. Thus, if the PBW basis elements are linearly independent in formula_185—which one can establish analytically—they must certainly be linearly independent in formula_68. (And, at this point, the isomorphism of formula_185 with formula_68 is apparent.)