Let G be a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p[ges ]0. In this paper, we classify all pairs (X, Y) of reductive subgroups of G which have a dense (X, Y)-double coset in G. In fact, we show that there is a dense (X, Y)-double coset in G precisely when G=XY is a factorisation. The possible factorisations that can occur have recently been determined in [11, Theorem A].

We now state our main result.