Smooth Cartan triples and Lie twists over Hausdorff \'etale Lie groupoids

We describe how to recover a Lie structure on a twist over a Hausdorff \'etale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff \'etale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We use these results in the setting of twists over \'etale groupoids to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections.