Minimal prime ideals in enveloping algebras of Lie superalgebras

Let ${\mathfrak g}$ be a finite dimensional Lie superalgebra over a field of characteristic zero. Let $U({\mathfrak g})$ be the enveloping algebra of ${\mathfrak g}$. We show that when ${\mathfrak g} = b(n)$, then $U({\mathfrak g})$ is not semiprime, but it has a unique minimal prime ideal; it follows then that when ${\mathfrak g}$ is classically simple, $U({\mathfrak g})$ has a unique minimal prime ideal. We further show that when ${\mathfrak g}$ is a finite dimensional nilpotent Lie superalgebra, then $U({\mathfrak g})$ has a unique minimal prime ideal.