Reduced forms of linear differential systems and the intrinsic Galois-Lie algebra of Katz

13 pages; International audience; Generalizing the main result of (Aparicio, Compoint, Weil 2013), we prove that a system is in reduced form in the sense of Kochin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement, which was implicit in (Aparicio, Compoint, Weil 2013) and is a crucial ingredient of (Barkatou, Cluzeau, Di Vizio, Weil 2016). We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group, which is actually another fundamental ingredient of the algorithm in (Barkatou, Cluzeau, Di Vizio, Weil 2016).