When G is a non-reductive connected affine algebraic group with dim R(G)>0, G can be embedded as the unit group of an irreducible affine algebraic monoid M. We also assume that G is a nonnilpotent group. Most of time, M has infinitely many minimal idempotents, all lying in the kernel of M (two-sided semigroup-theoretic minimal ideal of M), ker(M). Conversely, all idempotents in ker(M) are minimal. In this case, the one-sided, two-sided centralized of an idempotent e\in ker(M) in G, Z_G^l(e), Z_G^r(e), Z_G(e), all all proper connected closed non-normal subgroups of G. Let H be such a subgroup og G. Since R_u(H)<R_u(G) (ref: W. Huang, Kernels, regularity, unipotent radicals in a linear algebraic monoids, Forum Math. 23(2011), 803-834.), by Grosshans LNM 1673, Theorem 7.1, the homogeneous space G/H is affine.
Wenxue Huang
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