The coexponential or symmetrization map is the unique linear map from the symmetric Hopf algebra on the (underlying vector space of a) Lie algebra to the enveloping Hopf algebra of
such that for every “linear” element . This map is also given by the formula
where are arbitrary and are the same elements understood in . The coexponential map is an isomorphism of coalgebras .
The term symmetrization map is more widely used but it is a bit imprecise as it is used also for more general coalgebra isomorphisms sometimes called also generalized symmetrization maps.