  We prove that a Banach algebra $B$ that is a completion of the universal
enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak{g}$
satisfies a polynomial identity if and only if the nilpotent radical
$\mathfrak{n}$ of $\mathfrak{g}$ is associatively nilpotent in $B$.
Furthermore, this holds if and only if a certain polynomial growth condition is
satisfied on $\mathfrak{n}$.
