Example

Let $G$ be a Lie group. We make $G \times G$ operate on $G$ on the left by $(g_1, g_2)x = g_1xg_2^{-1}$. Let $\rho$ be the orbital mapping of $e$. Then the restrictions of $T_{(e, e)}(\rho)$ to $T_{(e, e)}(G \times \{e\}) = T_e(G) \times \{0\}$ and to

are isomorphisms of these spaces onto $T_e(G)$. Hence $T_{(e, e)}(\rho)$ is surjective and $\operatorname{Ker} T_{(e, e)}(\rho)$ admits for example the topological supplement $T_e(G) \times \{0\}$ in $T_{(e, e)}(G \times G)$. Thus $\rho$ is a submersion at $(e, e)$. Hence $G$ is a left Lie homogeneous space of $G \times G$.