You will find what you describe in the first reference. This describes how the Bernouilli numbers arise when studying the universal enveloping algebra. I have seen unpublished notes on this from a talk by Kostant in the '70s. This is a strong form of the PBW theorem and is closer to Poincare's result. This is discussed in the second reference. This is an early version of universal quantisation.

MR2301242 (2008d:17015)  Durov, Nikolai ;  Meljanac, Stjepan ;  Samsarov, Andjelo ;  Škoda, Zoran . A universal formula for representing Lie algebra generators as formal
 power series with coefficients in the Weyl algebra.
 J. Algebra  309  (2007),  no. 1, 318--359.

MR1793103 (2001f:01039)  Ton-That, Tuong ;  Tran, Thai-Duong . Poincaré's proof of the so-called Birkhoff-Witt theorem.
 Rev. Histoire Math.  5  (1999),  no. 2, 249--284 (2000).