Let $U:=\{A:=U_0 \subseteq U_1 \subseteq \cdots \subseteq U_i\}$ be an $\mathbb{N}$-filtered associative ring.
Definition: We say $U,U_i$ is an "almost commutative PBW-algebra if the following holds: $k \rightarrow A$ is a $k$-algebra ($k$ a commutative unital ring) with $k$ in the center of $U$. $U$ is an associative unital ring with multiplication inducing a map
$$U_i \times U_j \rightarrow U_{i+j}.$$ The associated graded ring $Gr(U,U_i)$ is commutative. Let $V:=U_1/U_0$. There is a canonical map $\rho: Sym_A^*(V) \rightarrow Gr(U,U_i)$ and $\rho$ is an isomorphism of graded $A$-algebras.
Example: If $k$ is a field and $\mathfrak{g}$ is a $k$-Lie algebra with $f\in Z^2(\mathfrak{g},k)$ is a 2-cocycle we may define the "Sridharan enveloping algebra" $U:=U(\mathfrak{g},f)$. The associative ring $U$ has a filtration $U_i$ and there is a canonical isomorphism of graded algebras $\rho: Sym_k^*(\mathfrak{g}) \cong Gr(U,U_i)$. If $f=0$ we get the universal enveloping algebra $U(\mathfrak{g})$.
Question: Has the center $Z(U)$ for a PBW algebra $U$ been calculated? What about the Sridharan enveloping algebra $Z(U(\mathfrak{g},f))$?
If $k$ is the complex number field and $\mathfrak{g}_s$ is a semi simple Lie algebra, it follows the center $Z(U(\mathfrak{g}_s))$ is known by the Harish-Chandra isomorphism. If $L:=\mathfrak{g}_{sv} \rtimes \mathfrak{g}_s$ where $\mathfrak{g}_{sv}$ is solvable, has $Z(U(L)) \subseteq U(L)$ been calculated? I ask for a reference to the litterature.