TITLE: Nice proofs of the Poincaré–Birkhoff–Witt theorem
QUESTION [48 upvotes]: Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$, with an ordered basis $x_1 < x_2 < ... < x_n$.
We define the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ to be the free noncommutative algebra $k\langle x_1,...,x_n\rangle$ modulo the relations $(x_ix_j - x_jx_i = [x_i,x_j])$.
The Poincaré–Birkhoff–Witt (PBW) theorem states that $U(\mathfrak{g})$ has a basis consisting of lexicographically ordered monomials i.e. monomials of the form $x_1^{e_1}x_2^{e_2}...x_n^{e_n}$. Checking that this basis spans $U(\mathfrak{g})$ is trivial, so the work lies in showing that these monomials are linearly independent.
One standard proof of PBW is to construct a $\mathfrak{g}$-action on the commutative polynomial ring $k[y_1,...,y_n]$ by setting $x_1^{e_1}x_2^{e_2}...x_n^{e_n}\cdot 1 = y_1^{e_1}y_2^{e_2}...y_n^{e_n}$ and verify algebraically that this gives rise to a well-defined representation of $\mathfrak{g}$. Details can be found in Dixmier's book on enveloping algebras.
What other proofs of PBW are there out there?
Are there nice reformulations of the above proof from a different perspective, such as one that emphasizes the universal property of $U(\mathfrak{g})$?
However, I would be especially interested in learning about proofs which are not just repackaged versions of the same algebraic manipulations used in the above proof (for example, geometric proofs which appeal to some property of $U(\mathfrak{g})$ as differential operators, etc.). If we allow ourselves more tools than just plain algebra, what other proofs of PBW can we get?

REPLY [5 votes]: Recently, me and Vladimir Dotsenko formulated PBW in terms of maps of monads in https://arxiv.org/abs/1804.06485. As a by product, one gets the classical PBW theorem is the manifestation that, in characteristic zero, the associative operad is free as a right Lie module through the map $\phi:\mathsf{Lie} \to\mathsf{Ass}$ that sends the bracket to the commutator. 
The left adjoint to the forgetful map from associative algebras to Lie algebras is the enveloping functor, and since $\mathsf{Ass} = \mathsf{Com}\circ\mathsf{Lie}$, one gets the desired natural isomorphism between $\phi_!(\mathfrak g) = U(\mathfrak g)$ and $\mathsf{Com}(\mathfrak g) = S(g)$. 
To prove that $\phi$ is free, one observes that we're dealing with weight graded connected operads $P$, so we can prove a weight graded module $M$ is free by showing that the bar complex $B(M,P,1)$ is acyclic (i.e. concentrated in degree 0). 
Since freeness is an homological statement, one can use homological methods to prove this: we filter $\mathsf{Ass}$ by the powers of the ideal generated by the Lie bracket, and note that the associated graded object that results is the Poisson operad, which is $\mathsf{Com}\circ\mathsf{Lie}$, giving the result.