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Cartier-KonstantKostant-Milnor-Moore theorem

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Cartier-Konstant-Milnor-Moore theorem

If $k$ is an algebraically closed field of characteristic zero and $H$ is a cocommutative Hopf algebra, then $$ H \cong U(P(H)) \ltimes kG(H). $$ What happens if the field is not algebraically closed? Is the theorem still true or is there any counterexample? What about characteristic different from zero?