Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ be its universal enveloping algebra, that is $$U(\mathfrak{g}) = \bigotimes\mathfrak{g}/\mathcal{K},$$ where $\bigotimes\mathfrak g$ denotes the tensor algebra of $\mathfrak g$ and $\mathcal K$ is the ideal of $\bigotimes\mathfrak g$ generated by the elements of the form $[x,y]-x\otimes y+y\otimes x$ with $x,y\in\mathfrak g$.
As far as I understand it, if $\mathfrak g$ is the Lie algebra of a Lie group $G$, and therefore we can regard $\mathfrak g$ as the set of left-invariant vector fields in $G$, the Poincaré-Birkoff-Witt theorem tells us that $U(\mathfrak g)$ corresponds the set of left-invariant differential operators of all orders on $G$. This means that, given $h\in G$, if we denote the left and right translations on $G$ by $$\lambda(h)f(g) = f(h^{-1}g),\, \rho(h)f(g) = f(gh),\quad\forall g\in G,$$ then $$\lambda(h)\circ D = D\circ\lambda(h),\quad\forall D\in U(\mathfrak g).$$
We denote by $Z(U(\mathfrak g))$ the center of $U(\mathfrak g)$.
Now, to the main question: I have seen it mentioned that $$D\in Z(U(\mathfrak g)) \Leftrightarrow \rho(h)\circ D = D\circ \rho(h),$$ but I simply cannot understand how commutators and right translations are related. I would be grateful if someone could offer me some insight.
