Primitive element of a Hopf algebra ($\Delta x=x\otimes 1+1\otimes x$)

An element $x$ of positive degree is primitive when $\Delta x=x\otimes 1+1\otimes x$. Graded commutators of primitives are primitive, so they form a graded Lie algebra. In a universal enveloping algebra over $\mathbb{Q}$, the primitive elements of a Hopf algebra are exactly its given Lie algebra: the associated graded algebra of the word-length filtration is symmetric, where no nonzero homogeneous polynomial of length greater than one is additive under the diagonal.