Sentence:  A <WSD>"primitive element</WSD>" of a Hopf algebra is an element h\in H such that / :: \Delta h = 1\otimes h + h\otimes 1. / It is easily seen that the bracket [x,y] := xy - yx of two <WSD>primitive elements</WSD> is again a <WSD>primitive element</WSD>. It follows that <WSD>primitive elements</WSD> form a Lie algebra. For H = U(\mathfrak g) any element of \mathfrak g is primitive and in fact using the "Poincaré-Birkhoff-Win" theorem, one can show that the set of <WSD>primitive elements</WSD> of U(\mathfrak g) coincides with the Lie algebra \mathfrak g. 
Meaning of the word marked with <WSD> tags:  An element "x" ∈ "C" such that "μ"("x") = "x" &#X2297; "g + g" &#X2297; "x", where "μ" is the comultiplication and "g" is an element that maps to the multiplicative identity 1 of the base field under the counit (in particular, if "C" is a bialgebra, "g" = 1).