Gauss Decomposition with Prescribed Semisimple Part: Short Proof

We give a uniform short proof of the fact that the intersection of every non-central conjugacy class in a Chevalley group and a big Gauss cell is non-empty and that this intersection contains elements with any prescribed semisimple part. The interest in this property stems at least in part from its relation to Oreʹs and Thompsonʹs conjectures in the theory of finite groups.