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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 smoothstructure on the unit space of a Hausdorff \'etale groupoid can be extended toa Lie-groupoid structure on the whole groupoid. We introduce Lie twists overHausdorff Lie groupoids, building on Kumjian's notion of a twist over atopological groupoid. We establish necessary and sufficient conditions on afamily of sections of a twist over a Lie groupoid under which the twist can bemade into a Lie twist so that all the specified sections are smooth. We usethese results in the setting of twists over \'etale groupoids to describeconditions on a Cartan pair of C*-algebras and a family of normalisers of thesubalgebra, under which Renault's Weyl twist for the pair can be made into aLie twist for which the given normalisers correspond to smooth sections.

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