  The theory of generalized Nijenhuis torsions, which extends the classical
notions due to Nijenhuis and Haantjes, offers new tools for the study of normal
forms of operator fields. We propose a general result ensuring that, given a
family of commuting operator fields whose generalized Nijenhuis torsion of
level $l$ vanishes, there exists a local chart where all operators can be
simultaneously block-diagonalized. We also introduce the notion of generalized
Haantjes algebra, consisting of operators with a vanishing higher-level
torsion, as a new algebraic structure naturally generalizing standard Haantjes
algebras.
