Images are immersed Lie subgroups

Give $B=G\mathrm{/}K$ the quotient topology and let $q:G\to B$ be the coset map. It is open because $q^{-1}(q(O))=OK$ is a union of right translates of an open set $O$. It is Hausdorff: the equivalence relation is the closed set $R=\{(g,h):g^{-1}h\in K\}$, and if $(g,h)\mathrm{\notin }R$, a product neighborhood $U\times V$ disjoint from $R$ gives disjoint open quotient neighborhoods $q(U)$ and $q(V)$. Images under the open map $q$ of a countable basis of $G$ form a countable basis of $B$.

In a constant-rank product chart from [F1], choose the transverse slice $S$ obtained by setting the kernel coordinates to zero. The restriction $q{\mathrm{\mid }}_S$ is bijective onto $q(U)$: points have the same $F$-value exactly when they differ by an element of $K$, and the normal form makes each local fibre meet $S$ once. It is a homeomorphism because an open subset of $S$ thickens in the kernel coordinates to an open subset of $U$ with the same $q$-image. These charts make $B$ a Hausdorff second-countable smooth manifold and make $q$ locally the projection $(u,v)\mapsto u$, hence a surjective submersion. Their changes are smooth because each has the smooth local section supplied by its slice.

Normality of $K$ gives $B$ its quotient group law. Multiplication and inversion are smooth: near any arguments, choose the smooth local sections from step 2.1 and express the descended maps as $q(s_1(x)s_2(y))$ and $q(s(x{)}^{-1})$. Thus $B$ is a Lie group and $q$ is a smooth homomorphism.

Define $j:B\to H$ by $j(gK)=F(g)$. Algebraically this is a well-defined injective homomorphism with image $\mathrm{im}F$. In the local coordinates of step 2.1 and the target constant-rank chart, $j$ is $u\mapsto (u,0)$, so it is a smooth immersion. Therefore $j(B)$ with the transported intrinsic structure is an immersed Lie subgroup, and $F=j\circ q$.

At the identity, $d{F}_e=d{j}_{eK}\circ d{q}_e$. The differential $d{q}_e$ is surjective with kernel $T_{e}K=\ker d{F}_e$ by the local projection and [A1], while $d{j}_{eK}$ is injective. Hence $d{j}_{eK}(T_{eK}B)=\mathrm{im}d{F}_e$, which is the tangent algebra of the immersed image.

If another manifold structure on the same image makes the corestriction from $G$ a surjective submersion, its local smooth sections show that the identity map in either direction is locally a composite of that corestriction with a local section for the other structure. Thus the identity is a diffeomorphism and the structure is unique. Rank zero, trivial image, noninjective $F$, and disconnected groups are included. Nothing in the construction identifies the intrinsic topology with the subspace topology of $H$; no embeddedness or closedness conclusion is asserted. Choice is inherited only through [A1].