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We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locallynilpotent radical of a splitting Borel subalgebra of a simple complex Liealgebra $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$,$\mathfrak{so}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$.There are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$,and we provide explicit generators of the center of $U(\mathfrak{n})$ in allcases. 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 apreliminary result, we provide a characterization of the centrally generatedprimitive ideals in the enveloping algebra of the nilradical of a Borelsubalgebra of $\mathfrak{sl}_n(\mathbb{C})$, $\mathfrak{sp}_{2n}(\mathbb{C})$.

Infinite Kostant cascades and centrally generated primitive ideals of $U(\mathfrak{n})$ in types $A_{\infty}$, $C_{\infty}$