Poincaré–Birkhoff–Witt theorem

By [L1] and [L2], the ordered monomials are simultaneously spanning and linearly independent, hence form a basis of $U(\mathfrak{g})$.

To verify the reverse implication in the stated equivalence, suppose that $\sigma$ is a graded-algebra isomorphism. The ordered commutative monomials form a basis of $S(\mathfrak{g})$ by [L3]. For spanning, use induction on $n$: if $u\in F_{n}U(\mathfrak{g})$, surjectivity of ${\sigma }_n$ expresses its class modulo $F_{n-1}$ as a finite linear combination of the classes of ordered length-$n$ monomials. Subtracting the same combination in $U(\mathfrak{g})$ leaves an element of $F_{n-1}$, to which the induction hypothesis applies. For independence, take a finite relation among ordered monomials and let $n$ be its largest occurring length. Its degree-$n$ class is the image under the injective map ${\sigma }_n$ of the corresponding combination of distinct ordered commutative monomials, so all degree-$n$ coefficients vanish; descending induction eliminates the rest. Thus the graded isomorphism implies the ordered-monomial basis assertion as well, including degree zero and the empty product.

The degree-preserving straightening result [L1] shows that every element of $F_{n}U(\mathfrak{g})$ is spanned by ordered monomials of length at most $n$; their linear independence follows from [L2]. Hence they form a basis of $F_n$, and $F_n\mathrm{/}{F}_{n-1}$ has as a basis their classes of length exactly $n$.

In degree $n$, $\sigma$ sends each ordered commutative basis monomial from [L3] to the class of the identically ordered PBW monomial from step 2.1. It is therefore a bijection in every degree.

Since $\sigma$ is a graded algebra homomorphism by [L4] and is bijective on every graded component by step 3.1, it is a graded-algebra isomorphism. When $B$ is empty, both bases consist only of the empty monomial, so the boundary case is included.