Abstract. If G is a finite simple group of Lie type over a field containing more than 8 elements (for twisted groups l Xn(q l ) we require q > 8, except for 2 B 2 (q 2 ), 2 G 2 (q 2 ), and 2 F 4 (q 2 ), where we assume q 2 > 8), then G is the square of some conjugacy class and consequently every element in G is a commutator.

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Let G = G(K) where G is a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells BẇB of G with conjugacy classes C of G, in particular, we consider the question, when is BẇB ∩ C = ∅.

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.