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.