Let $G$ be a smooth Lie group with Lie algebra $\mathfrak g$. The adjoint action of $G$ on $\mathfrak g$ extends to the complexification $\mathfrak g_{\mathbb C}$ and then to the universal enveloping algebra $U(\mathfrak g_{\mathbb C})$.
The center $\mathfrak z$ of $U(\mathfrak g_{\mathbb C})$ consists of those elements which are invariant under the $\operatorname{ad}$ action of $\mathfrak g$. Equivalently, $D$ is in the center if $\operatorname{Ad}(\exp(X))D = D$ for all $X \in \mathfrak g$.
Since the connected component $G^0$ of $G$ is generated by the image of $\exp(\mathfrak g)$, we can say that $D$ is in the center if and only if $\operatorname{Ad}(g)D = D$ for all $g \in G^0$.
When $G$ is not connected, it possible to have a central element of $U(\mathfrak g_{\mathbb C})$ which is not $\operatorname{Ad}$-invariant? I would guess the answer is yes, since $G$ could be the semidirect product of $G^0$ by some exotic group of automorphisms.