  We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locally
nilpotent radical of a splitting Borel subalgebra of a simple complex Lie
algebra $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$,
$\mathfrak{so}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$.
There are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$,
and we provide explicit generators of the center of $U(\mathfrak{n})$ in all
cases. We then fix $\mathfrak{n}$ with "largest possible" center of
$U(\mathfrak{n})$ and characterize the centrally generated primitive ideals of
$U(\mathfrak{n})$ for $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$,
$\mathfrak{sp}_{\infty}(\mathbb{C})$ in terms of the above generators. As a
preliminary result, we provide a characterization of the centrally generated
primitive ideals in the enveloping algebra of the nilradical of a Borel
subalgebra of $\mathfrak{sl}_n(\mathbb{C})$, $\mathfrak{sp}_{2n}(\mathbb{C})$.
