Enhanced orbit embedding

Let $\tilde{G}$ be an algebraic group acting on a variety $\tilde{L}$, and $G \subset \tilde{G}$ a subgroup which leaves a subvariety $L \subset \tilde{L}$ stable. For a $G$-orbit $O_G = G u (u \in L)$ in $L$, we can associate an orbit $O_{\tilde{G}} = \tilde{G} u$ of $\tilde{G}$ so that we get a map $L/G \to \tilde{L}/\tilde{G}$ between orbit spaces, though this map is usually not injective. In this note, when $G$ is a symmetric subgroup arising from an involutive anti-automorphism, we give certain sufficient conditions for the map $L/G \to \tilde{L}/\tilde{G}$ to be injective after the method of Ohta (2008). Our main concern here is to produce examples of enhanced Lie algebras (or enhanced $\theta$-representations). We also analyze an obstruction which prevents the orbit space inclusion.