  Let G be a connected linear algebraic group over an algebraically closed
field k, and let H be a connected closed subgroup of G. We prove that the
homogeneous variety G/H is a rational variety over k whenever H is solvable, or
when dim(G/H) < 11 and characteristic(k) = 0. When H is of maximal rank in G,
we also prove that G/H is rational if the maximal semisimple quotient of G is
isogenous to a product of almost-simple groups of type A, type C (when
characteristic(k) is not 2), or type B_3 or G_2 (when characteristic(k) = 0).
