Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let fraktur small g sign be its Lie algebra. Let k(G), respectively, k(fraktur small g sign), be the field of k-rational functions on G, respectively, fraktur small g sign. The conjugation action of G on itself induces the adjoint action of G on fraktur small g sign. We investigate the question whether or not the field extensions k(G)/k(G) ^G and k(fraktur small g sign)/k(fraktur small g sign)^G are purely transcendental. We show that the answer is the same for k(G)/k(G) ^G and k(fraktur small g sign)/k(fraktur small g sign)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C _n, and negative for groups of other types, except possibly G ₂. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.