Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?

Question

Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})$ has a quasi-triangular topological quasi-bialgebra structure. According to Theorem XIX 4.3 of Kassel's Quantum groups, for a quantum enveloping algebra $U_\hbar(\mathfrak{g})$ of $\mathfrak{g}$, there exists a gauge transformation $F$ exists and $(U(\mathfrak{g})[[\hbar]])_F$ and $U_\hbar(\mathfrak{g})$ are isomorphic. Is it possible to explicitly describe this gauge transformation F ? I would like to know just in the case of $\mathfrak{sl}_2$, not in the case of $\mathfrak{g}$ in general.