On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture

We explore the Jordan-Chevalley decomposition problem for an operator field in small dimensions. In dimensions three and four, we find tensorial conditions for an operator field $L$, similar to a nilpotent Jordan block, to possess local coordinates in which $L$ takes a strictly upper triangular form. We prove the Tempesta-Tondo conjecture for higher order brackets of Frölicher-Nijenhuis type.