= Graded Poincaré–Birkhoff–Witt theorem
{title2=$\operatorname{gr}U(L)\cong\operatorname{Sym}(L)$}
For a <graded Lie algebra> over <characteristic zero>, the associated graded algebra of its <universal enveloping algebra> under the word-length filtration is its <graded symmetric algebra>. Ordered homogeneous monomials form a <basis>, with exponent at most one on each odd Lie <basis> element. The defining graded commutation relations reorder every word, and the Jacobi identity resolves the triple-reordering ambiguities. In particular the <primitive elements of a Hopf algebra> of $U(L)$ are precisely $L$: in its associated graded algebra, a primitive polynomial of symmetric length greater than one would have a nonzero mixed term under the diagonal, which is impossible in <characteristic zero>.
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