One of my favorite facts about Lie groups is that Lie subgroups are weakly embedded (aka initial submanifolds). I'm trying to see how one can generalize this result.
"Conjecture": Let $G$ be a Lie group and $H, J$ two Lie subgroups s.t. $J \cap H = \{e\}$. Does this imply that $m: H \times J \to G$ given by multiplication is a weak embedding?
What I have gotten so far is that if $J$ is closed, $H$ embeds weakly into the homogeneous space $G/J$:
$H$ has Lie algebra $\mathfrak{h}$ and is an integral manifold of the distribution $\Gamma(g) = {dL_g}(\mathfrak{h})$. Since this is left-invariant, it "pushes down" to a distribution $\pi_* \Gamma \subset T(G/J)$, where $\pi : G \to G/J$ is the projection map. $H \to G/J$ has full rank and is injective, so it is an immersed submanifold and it also is a integral submanifold of $\pi_* \Gamma$. Hence $\pi_*\Gamma$ is involutive, so $H$ is weakly embedded because it is an integral manifold of an involutive distribution.
This is implied by the "conjecture": Consider a (not necessarily smooth/continuous) map $f: X \to H$ s.t. $\pi \circ i_H \circ f : X \to G/J$ is smooth ($i_H : H \to X$ is the inclusion). Locally the latter lifts to a smooth map $g : X \to G$ which must have values in $m(H \times J)$. Hence, (here we use the "conjecture"), $g$ factors as $g = i_{H \times J} \circ h$, so $$ \pi \circ i_H \circ f = \pi \circ i_{H \times J} \circ h = \pi \circ i_{H} \circ \mathit{pr}_1 \circ h $$ (where $\mathit{pr}_1 : H \times J \to H$ is the canonical projection). Since $\pi \circ i_H$ is injective, we obtain $f = \mathit{pr}_1 \circ h$, which is smooth.