Let g \mathfrak {g} be a finite-dimensional complex Lie algebra, and let U ( g ) U(\mathfrak {g}) be its universal enveloping algebra. We prove that if U ^ ( g ) \widehat {U}(\mathfrak {g}) , the Arens-Michael envelope of U ( g ) U(\mathfrak {g}) is stably flat over U ( g ) U(\mathfrak {g}) (i.e., if the canonical homomorphism U ( g ) → U ^ ( g ) U(\mathfrak {g})\to \widehat {U}(\mathfrak {g}) is a localization in the sense of Taylor (1972), then g \mathfrak {g} is solvable. To this end, given a cocommutative Hopf algebra H H and an H H -module algebra A A , we explicitly describe the Arens-Michael envelope of the smash product A # H A\# H as an “analytic smash product” of their completions w.r.t. certain families of seminorms.