Does anyone know if the following holds?

Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.

Note that if G is simply-connected then the centralizer of any element is connected. This implies that both elements lie in a torus and hence the conjecture holds.

On the other hand, the rank assumption is important since the diagonal orthogonal matrices D(-1,1,-1) and D(1,-1,-1) commute in SO(3,C) but one can prove that they don't belong to the same parabolic subgroup.

Does anyone know if the following holds?

Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.

To make things easier, you can assume that these elements are semi-simple.

Note that if G is simply-connected then the centralizer of any semi-simple element is connected. This implies that if both are semi-simple then they lie in a torus and hence the conjecture holds.

On the other hand, the rank assumption is important since the diagonal orthogonal matrices D(-1,1,-1) and D(1,-1,-1) commute in SO(3,C) but one can prove that they don't belong to the same parabolic subgroup.