Arens-Michael enveloping algebras and analytic smash products

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra, and let $U(\mathfrak{g})$ be its universal enveloping algebra. We prove that if $\widehat{U}(\mathfrak{g})$, the Arens-Michael envelope of $U(\mathfrak{g})$ is stably flat over $U(\mathfrak{g})$ (i.e., if the canonical homomorphism $U(\mathfrak{g})\to\widehat{U}(\mathfrak{g})$ is a localization in the sense of Taylor (1972), then $\mathfrak{g}$ is solvable. To this end, given a cocommutative Hopf algebra $H$ and an $H$-module algebra $A$, we explicitly describe the Arens-Michael envelope of the smash product $A\char93 H$ as an ``analytic smash product'' of their completions w.r.t. certain families of seminorms.