The universal enveloping algebra of a Lie algebra naturally becomes a Hopf algebra in which all the elements of the original Lie algebra are primitive elements. The Milnor-Moore theorem states conditions under which, conversely, any Hopf algebra generated by primitive elements is the universal enveloping algebra of the Lie algebra structure on these elements.

References

The result is due to

John Milnor, John Moore, On the structure of Hopf algebras, Annals of Math. 81 (1965), 211-264. (pdf)