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

Let G be a split simple group of type G₂ over a field k, and let g be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavskiĭ, V.L. Popov, and Z. Reichstein, we show that the function field k (g) is generated by algebraically independent elements over the field of adjoint invariants k^{(g) G}.