Proposition 25

Let G be a Lie group and $E(G)$ the enveloping algebra of $L(G)$. The canonical injection of $L(G)$ into $E(G)$ defines a homomorphism $\theta$ of the algebra $E(G)$ into the algebra $U(G)$. If K is of characteristic 0, $\eta$ is a bigebra isomorphism.

The bigebra $U(G)$ is cocommutative (Differentiable and Analytic Manifolds, R, 13.5.1) and the filtration $(U_s(G))$ is compatible with the bigebra structure. The set of primitive elements of $U(G)$ is $L(G)$. It then suffices to apply chapter II, § 1, no. 6, Theorem 1.

When K is of characteristic 0 we shall in future identify $U(G)$ with the enveloping algebra of $L(G)$. By (2) and Proposition 7 (ii), the mapping $t \mapsto t^\vee$ of $U(G)$ into $U(G)$ is then identified with the principal antiautomorphism of $U(G)$ (Chapter I, § 2, no. 4).