What are some nice corollaries or applications of the PBW theorem? There's this immediate corollary that a Lie algebra sits inside the universal enveloping algebra so in particular, the Lie algebra structure comes from an associative algebra. Why else is the theorem interesting?

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    $\begingroup$ For those who will wonder, "PBW" stands for Poincaré–Birkhoff–Witt. (And not, as my adviser sometimes claimed, "Peanut Butter & Witt".) $\endgroup$ – FredH Mar 24 at 0:08
  • $\begingroup$ Most proofs of Ado's theorem make use of the PBW theorem. $\endgroup$ – YCor Mar 24 at 20:05
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    $\begingroup$ MO crosspost: mathoverflow.net/questions/326233/… $\endgroup$ – YCor Mar 24 at 20:13
  • $\begingroup$ Analogues of the PBW theorem for other algebras can help in combinatorics. For example, the PBW theorem for the Weyl algebra in two variables $x$ and $\partial$ can be used to prove identities between Stirling numbers, because these Stirling numbers appear in the change-of-basis matrices from one basis of the Weyl algebra into another. (And the PBW theorem guarantees that the bases are indeed bases, thus allowing the comparison of coefficients.) Similarly, PBW theorems for quantum groups allow the construction of crystals. I can't think of a specific application of Lie algebra PBW, though. $\endgroup$ – darij grinberg Mar 24 at 20:24

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