Poincaré–Birkhoff–Witt theorem

Remarks

That the ordered monomials span is easy: any product of the $\iota(x_i)$ can be reordered. Each swap of neighbours changes it by a product of lower length, by the relation $xy - yx = [x, y]$. Linear independence is the hard part. A standard proof builds an action of $\mathfrak g$ on the polynomial ring $S(\mathfrak g)$ that sends each ordered monomial applied to $1$ to the matching polynomial monomial. The Jacobi identity is used to check that this is an action. In particular $\iota$ is injective, so every Lie algebra over a field is a Lie subalgebra of an associative algebra with the commutator bracket. Equivalently, the associated graded algebra of $U(\mathfrak g)$ for its filtration by length of products is isomorphic to the symmetric algebra $S(\mathfrak g)$. Example: for $\mathfrak{sl}_2$ with basis $f < h < e$, the elements $f^a h^b e^c$ with $a, b, c \ge 0$ form a basis of $U(\mathfrak{sl}_2)$. Over a commutative ring instead of a field, the theorem holds when $\mathfrak g$ is free as a module, but it can fail in general.

Going deeper

For a student

The theorem is the basic structural fact about $U(\mathfrak g)$. It shows that $\mathfrak g$ embeds in $U(\mathfrak g)$, that $U(\mathfrak g)$ has no zero divisors, and that $U(\mathfrak g)$ is Noetherian when $\mathfrak g$ is finite-dimensional, by passing to the polynomial ring $S(\mathfrak g)$. For a subalgebra $\mathfrak h \subseteq \mathfrak g$, it shows that $U(\mathfrak g)$ is a free $U(\mathfrak h)$-module, which is the key input for Verma modules and induced representations. It also gives the vector space isomorphism $S(\mathfrak g) \to U(\mathfrak g)$ by symmetrisation in characteristic $0$.