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The nicest one I have ever seen uses a mix of universal algebra and combinatorial algebra, and was given by P. J. Higgins in Baer Invariants and the Birkhoff-Witt Theorem, Journal of Algebra 11, pp. 469-482 (1969). I can send the 1969) (free PDF to anyone interestedlinked).

Then there is the purely computational one which works over any $\mathbb Q$-algebra as base "field" and was given in the book by Deligne-Morgan. See http://mathoverflow.net/questions/66683 for details.

Emanuela Petracci gave in her thesis another computational proof, which uses the language of bialgebras to make the manipulations manageable.

There is also Cohn's A remark on the Poincaré-Birkhoff-Witt Theorem, J. London Math. Soc. (1963) s1-38(1): 197-203. It also has a discussion topic on MO.

If $\mathfrak g$ is the Lie algebra of a Lie group over $\mathbb R$, then you can indeed prove PBW using geometry: see, e. g., Proposition 1.9 in PDF 1 of Chapter 2 of Helgason's Lie Groups lecture notes. However, I don't think it is realistic to use this as a general proof for PBW; Lie's Third Theorem seems to be hard and require PBW itself.

Poincaré might have proven PBW himself (at least over a field of characteristic $0$), but I don't understand his proof (at least in a modern translation, which might itself be erroneous).

I hate to say but the only of the above references that I found easily readable is Higgins's paper...

The nicest one I have ever seen uses a mix of universal algebra and combinatorial algebra, and was given by P. J. Higgins in Baer Invariants and the Birkhoff-Witt Theorem, Journal of Algebra 11, pp. 469-482 (1969). I can send the PDF to anyone interested.

Then there is the purely computational one which works over any $\mathbb Q$-algebra as base "field" and was given in the book by Deligne-Morgan. See http://mathoverflow.net/questions/66683 for details.

Emanuela Petracci gave in her thesis another computational proof, which uses the language of bialgebras to make the manipulations manageable.

There is also Cohn's A remark on the Poincaré-Birkhoff-Witt Theorem, J. London Math. Soc. (1963) s1-38(1): 197-203. It also has a discussion topic on MO.

If $\mathfrak g$ is the Lie algebra of a Lie group over $\mathbb R$, then you can indeed prove PBW using geometry: see, e. g., Proposition 1.9 in PDF 1 of Chapter 2 of Helgason's Lie Groups lecture notes. However, I don't think it is realistic to use this as a general proof for PBW; Lie's Third Theorem seems to be hard and require PBW itself.

Poincaré might have proven PBW himself (at least over a field of characteristic $0$), but I don't understand his proof (at least in a modern translation, which might itself be erroneous).

I hate to say but the only of the above references that I found easily readable is Higgins's paper...