It appears that pre-Bourbaki sources were not familiar with Poincaré's paper.īirkhoff and Witt do not mention Poincaré's work in their 1937 papers. She further says that the theorem was subsequently completely demonstrated by Witt and Birkhoff. Following this old tradition, Fofanova in her encyclopaedic entry says that Poincaré obtained the first variant of the theorem. They have found out that the majority of the sources before Bourbaki's 1960 book call it Birkhoff-Witt theorem. Ton-That and Tran have investigated the history of the theorem. 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. the General linear Lie algebra while Poincaré later stated it more generally in 1900. Extend h to all canonical monomials as follows: if ( x 1, x 2. , x n) of elements of X which is non-decreasing in the order ≤, that is, x 1 ≤ x 2 ≤. A canonical monomial over X is a finite sequence ( x 1, x 2. Let L be a Lie algebra over K and X a totally ordered basis of L. If L is a Lie algebra over a field K, let h denote the canonical K- linear map from L into the universal enveloping algebra U( L). In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases of which the elements are totally ordered by some relation which we denote ≤. Recall that any vector space V over a field has a basis this is a set S such that any element of V is a unique (finite) linear combination of elements of S.