While the OP asked for a more geometric rather than algebraic proof, I would like to point out Bergman's very nice paper,

The diamond lemma for ring theory, Advances in Mathematics 29 (1978) 178-218.

Here's a link to the paper (thanks to Darij for pointing this out!):

http://www.sciencedirect.com/science/article/pii/0001870878900105

Also, there is this blog post by David Speyer which summarizes it, in the context of other "diamond lemma" results:

http://sbseminar.wordpress.com/2009/11/20/the-diamond-lemma/

Essentially, for algebras whose defining relations fit into a framework of "find a certain type of monomial, and replace it with a linear cominbation of simpler ones", for some suitable notion of "simple", there are obvious obstructions to linear independence of reduced expressions: one may have a monomial (AB)C=A(BC), such that both AB and BC can be reduced. In that case, the ambiguity is called resolvable if upon further simplifying to reduced expressions, the two different expressions yield the same result. Of course if they didn't, then it would mean there was a linear relation amongst reduced monomials.

Bergman's result says that if you can resolve such simple overlap ambiguities, featuring just one overlap (and also something called inclusion ambiguity - which are rarer, and can always be weeded out anyways, as he explains), then the PBW monomials indeed form a basis. In the case of Lie algebras, checking this reduces to nothing more than the Jacobi identity, as he shows.

I really like the approach via Diamond Lemma as it is very elementary to prove, and the conditions of the Lemma are easy to verify in practice. It is general enough to apply to very many "almost" commutative algebras, such as arise in quantum groups and related areas.

While the OP asked for a more geometric rather than algebraic proof, I would like to point out Bergman's very nice paper,

The diamond lemma for ring theory, Advances in Mathematics 29 (1978) 178-218.

Here's a link to the paper (thanks to Darij for pointing this out!):

https://doi.org/10.1016/0001-8708(78)90010-5

Also, there is this blog post by David Speyer which summarizes it, in the context of other "diamond lemma" results:

https://sbseminar.wordpress.com/2009/11/20/the-diamond-lemma/

Essentially, for algebras whose defining relations fit into a framework of "find a certain type of monomial, and replace it with a linear cominbation of simpler ones", for some suitable notion of "simple", there are obvious obstructions to linear independence of reduced expressions: one may have a monomial (AB)C=A(BC), such that both AB and BC can be reduced. In that case, the ambiguity is called resolvable if upon further simplifying to reduced expressions, the two different expressions yield the same result. Of course if they didn't, then it would mean there was a linear relation amongst reduced monomials.

Bergman's result says that if you can resolve such simple overlap ambiguities, featuring just one overlap (and also something called inclusion ambiguity - which are rarer, and can always be weeded out anyways, as he explains), then the PBW monomials indeed form a basis. In the case of Lie algebras, checking this reduces to nothing more than the Jacobi identity, as he shows.

I really like the approach via Diamond Lemma as it is very elementary to prove, and the conditions of the Lemma are easy to verify in practice. It is general enough to apply to very many "almost" commutative algebras, such as arise in quantum groups and related areas.

While the OP asked for a more geometric rather than algebraic proof, I would like to point out Bergman's very nice paper,

The diamond lemma for ring theory, Advances in Mathematics 29 (1978) 178-218.

Sorry i don't have a link for it publicly, but there is this blog post by David Speyer which summarizes it, in the context of other "diamond lemma" results:

http://sbseminar.wordpress.com/2009/11/20/the-diamond-lemma/

Essentially, for algebras whose defining relations fit into a framework of "find a certain type of monomial, and replace it with a linear cominbation of simpler ones", for some suitable notion of "simple", there are obvious obstructions to linear independence of reduced expressions: one may have a monomial (AB)C=A(BC), such that both AB and BC can be reduced. In that case, the ambiguity is called resolvable if upon further simplifying to reduced expressions, the two different expressions yield the same result. Of course if they didn't, then it would mean there was a linear relation amongst reduced monomials.

Bergman's result says that if you can resolve such simple overlap ambiguities, featuring just one overlap (and also something called exclusion ambiguity), then the PBW monomials form a basis. In the case of Lie algebras, this reduces to nothing more than the Jacobi identity, as he shows.

I really like the approach via Diamond Lemma as it is very elementary to prove, and the conditions of the Lemma are easy to verify in practice. It is general enough to apply to very many "almost" commutative algebras, such as arise in quantum groups and related areas.

While the OP asked for a more geometric rather than algebraic proof, I would like to point out Bergman's very nice paper,

The diamond lemma for ring theory, Advances in Mathematics 29 (1978) 178-218.

Here's a link to the paper (thanks to Darij for pointing this out!):

http://www.sciencedirect.com/science/article/pii/0001870878900105

Also, there is this blog post by David Speyer which summarizes it, in the context of other "diamond lemma" results:

http://sbseminar.wordpress.com/2009/11/20/the-diamond-lemma/

Essentially, for algebras whose defining relations fit into a framework of "find a certain type of monomial, and replace it with a linear cominbation of simpler ones", for some suitable notion of "simple", there are obvious obstructions to linear independence of reduced expressions: one may have a monomial (AB)C=A(BC), such that both AB and BC can be reduced. In that case, the ambiguity is called resolvable if upon further simplifying to reduced expressions, the two different expressions yield the same result. Of course if they didn't, then it would mean there was a linear relation amongst reduced monomials.

Bergman's result says that if you can resolve such simple overlap ambiguities, featuring just one overlap (and also something called inclusion ambiguity - which are rarer, and can always be weeded out anyways, as he explains), then the PBW monomials indeed form a basis. In the case of Lie algebras, checking this reduces to nothing more than the Jacobi identity, as he shows.

I really like the approach via Diamond Lemma as it is very elementary to prove, and the conditions of the Lemma are easy to verify in practice. It is general enough to apply to very many "almost" commutative algebras, such as arise in quantum groups and related areas.