Knowing a country’s laws can be incredibly helpful for genealogy research. A law can tell us in what order a census was enumerated, who could have inherited, or what taxes your ancestor had to pay (and thus, what documents would have been created).
ANLux has digitized records that should make your search easier. A small number of “fonds” or “collections” of the Conseil d’Etat covering the period from 1856-1940 are available on their website at http://www.anlux.public.lu/fr/documents-numerises/photos/conseil-etat.html. This collection includes everything from the revisions to the electoral law to the function of the library. It’s a bit eclectic but provides a fun overview to your ancestor’s life.