Cartan subalgebras for non-principal twisted groupoid $C^{\ast }$-algebras

Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C_r^{\ast }(G,\mathrm{\Sigma })$ for any twist $\mathrm{\Sigma }$ over $G$. However, there are many groupoids which are not topologically principal, yet their (twisted) $C^{\ast }$-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle $c$ on $G$ and a subgroupoid $S\subseteq G$ under which $C_r^{\ast }(S,c)$ is Cartan in $C^*_r(G, c)$\mathrm{.}When$G$ is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.