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 proper connected closed non-normal subgroups of $G$. Let $H$ be such a subgroup of $G$. 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.

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 centralizers of an idempotent $e \in \ker(M)$ in $G$, $Z_G^l(e)$, $Z_G^r(e)$, $Z_G(e)$, all proper connected closed non-normal subgroups of $G$. Let $H$ be such a subgroup of $G$. 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.

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.

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 proper connected closed non-normal subgroups of $G$. Let $H$ be such a subgroup of $G$. 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.