The Lie algebra of type G2 is rational over its quotient by the adjoint action

Let G be a split simple group of type $G_2$ over a field k, and let $\mathfrak{g}$ be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavski?, V.L. Popov, and Z. Reichstein, we show that the function field $k(\mathfrak{g})$ is generated by algebraically independent elements over the field of adjoint invariants $k{(\mathfrak{g})}^G$.