History of the theorem[edit]
In four papers from the 1880s Alfredo Capelli proved, in different terminology, what is now known as the Poincaré–Birkhoff–Witt theorem in the case of the General linear Lie algebra; while Poincaré later stated it more generally in 1900.[2] Armand Borel says that these results of Capelli were "completely forgotten for almost a century", and he does not suggest that Poincaré was aware of Capelli's result.[2]
Ton-That and Tran [3] have investigated the history of the theorem. They have found out that the majority of the sources before Bourbaki's 1960 book call it Birkhoff-Witt theorem. Following this old tradition, Fofanova[4] in her encyclopaedic entry says that Poincaré obtained the first variant of the theorem. She further says that the theorem was subsequently completely demonstrated by Witt and Birkhoff. It appears that pre-Bourbaki sources were not familiar with Poincaré's paper.
Birkhoff [5] and Witt [6] do not mention Poincaré's work in their 1937 papers. Cartan and Eilenberg[7] call the theorem Poincaré-Witt Theorem and attribute the complete proof to Witt. Bourbaki[8] were the first to use all three names in their 1960 book. Knapp presents a clear illustration of the shifting tradition. In his 1986 book[9] he calls it Birkhoff-Witt Theorem, while in his later 1996 book[10] he switches to Poincaré-Birkhoff-Witt Theorem.
It is not clear whether Poincaré's result was complete. Ton-That and Tran[3] conclude that "Poincaré had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff". On the other hand, they point out that "Poincaré makes several statements without bothering to prove them". Their own proofs of all the steps are rather long according to their admission. Borel states that Poincaré "more or less proved the Poincaré-Birkhoff-Witt theorem" in 1900.[2]
