Intermediate subalgebras of Cartan embeddings in rings and C*-algebras

Let $D\subseteq A$ be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist $\Sigma \to G$ whose twisted Steinberg algebra is isomorphic to A in a way that preserves D. In this paper, we show there is a lattice isomorphism between wide open subgroupoids of G and subalgebras C such that $D\subseteq C\subseteq A$ and $D\subseteq C$ is a quasi-Cartan pair. We also characterize which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra C with $D\subseteq C\subseteq A$ has $D\subseteq C$ a diagonal/Cartan/quasi-Cartan pair. In the diagonal case, when the coefficient ring is a field, it is all of them. Beyond that, only pairs that are close to being diagonal have this property. We then apply our techniques to C*-algebraic inclusions and give a complete characterization of which Cartan pairs $D\subseteq A$ have the property that every C*-subalgebra C with $D\subseteq C\subseteq A$ has $D\subseteq C$ a Cartan pair.