Effective descent for differential operators

A theorem of N. Katz (1990) [Ka], , p. 45, states that an irreducible differential operator L over a suitable differential field k, which has an isotypical decomposition over the algebraic closure of k, is a tensor product L=M⊗kN of an absolutely irreducible operator M over k and an irreducible operator N over k having a finite differential Galois group. Using the existence of the tensor decomposition L=M⊗N, an algorithm is given in É. Compoint and J.-A. Weil (2004) [C-W], which computes an absolutely irreducible factor F of L over a finite extension of k. Here, an algorithmic approach to finding M and N is given, based on the knowledge of F. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields k which are C1-fields.