[@IgnatyevPenkov1 Theorem 3.1] that, for $\Phi=A_{n-1}$ or $\Phi=C_n$, $J$ is centrally generated if and only if $J=J(f)$ for a certain Kostant form $f$. In this paper, we prove that this fact is also true for $\Phi=B_n$ or $\Phi=D_n$.

More precisely, let $P_{\beta}$, $\beta\in{\mathcal{B}}$, be the set of canonical generators of $Z({\mathfrak{n}})$ (the explicit formulas for $P_{\beta}$ are given in Subsection \[sst:center\_fd\]). Let $J$ be a primitive ideal of $U({\mathfrak{n}})$. Since $J$ is the annihilator of a simple ${\mathfrak{n}}$-module, given $\beta\in{\mathcal{B}}$, there exists the unique $c_{\beta}\in{\mathbb{C}}$ such that $P_{\beta}-c_{\beta}\in J$. Our first main result (see Theorem \[theo:ideal\_fd\_Bn\_Dn\]) claims that the following conditions are equivalent: $$\begin{split}
&\text{\textup{i)} $J$ is centrally generated};\\
&\text{\textup{ii)} all scalars $c_{\beta}$, $\beta\in{\mathcal{B}}\setminus\Delta$, are nonzero};\\
&\text{\textup{iii)} $J=J(f)$ for a Kostant form $f$}.
\end{split}$$ If these conditions are satisfied, then we present an explicit way how to reconstruct $f$ by $J$.

Suppose now that ${\mathfrak{n}}$ is the locally nilpotent radical of a Borel subalgebra ${\mathfrak{b}}$ of the simple finitary infinite-dimensional complex Lie algebra ${\mathfrak{g}}={\mathfrak{so}}_{\infty}({\mathbb{C}})$. The center of the enveloping algebra $U({\mathfrak{n}})$ was described in [@IgnatyevPenkov1 Theorem 2.6]. There are infinitely many isomorphism classes of Lie algebras ${\mathfrak{n}}$, and we concentrate on the case when $U({\mathfrak{n}})$ has “largest possible” center, which 