The Lie-Theoretic Cost of Breaking the Jacobian Conjecture

Van den Essen and Hubbers characterized linear triangularizability of polynomial maps X + H by strong nilpotence of the Jacobian family. This note records the counterexample-side Lie-theoretic form of that boundary. Suppose every evaluated Jacobian matrix JH(a) is nilpotent, and define the differential Lie algebra g_H = Lie generated by {JH(a) : a in C^n}. Then g_H is solvable if and only if JH is strongly nilpotent, equivalently if and only if X + H is linearly triangularizable. Consequently, every homogeneous Keller counterexample has a non-solvable differential Lie algebra. By Levi decomposition, that algebra contains a subalgebra isomorphic to sl_2(C). The July 2026 counterexample to the Jacobian Conjecture makes this a structural statement about actual Keller counterexamples. The note also formulates the resulting canonicity problem: whether any semisimple Lie structure can be attached to the global nonproperness obstruction independently of the chosen homogeneous presentation. Version 0.1. Research note. Not peer reviewed.