Is the field of functions on the Lie algebra pure over the invariant subfield?

Let $G$ be a connected reductive algebraic group over a field $k$ of characteristric zero and let $\mathrm{Lie}G$ be its Lie algebra. Let $k(G)$ and $k(\mathrm{Lie}G)$ be the fields of rational functions on $G$ and $\mathrm{Lie}G$ respectively. The conjugation action of $G$ on itself and the adjoint action of $G$ on $\mathrm{Lie}G$ determine the invariant subfields $k(G)^G$ and $k(\mathrm{Lie}G)^G$ of $k(G)$ and $k(\mathrm{Lie}G)$ respectively. In the talk the following problems will be addressed and answered: Is the field extension $k(G)/k(G)^G$ pure (i.e., purely transcendental)? stably pure? The same questions for the field extension $k(\mathrm{Lie}G)/\mathrm{Lie}(G)^G$. These questions come up in connection with the counterexamples to the Gelfand–Kirillov conjecture and they are also naturally related to the birational counterpart of the classical classification problem of representations with free module of covariants. The talk is based on the joint results with J.-L. Colliot-Thélène, B. Kunyavskii, and Z. Reichstein.