Engel's theorem

Engel's theorem is a concept in algebra that helps scientists identify a special type of mathematical structure known as a " nilpotent Lie algebra." It states that if, for every element in this structure, repeatedly applying a certain process (called the adjoint action) eventually results in zero, then the entire structure is “nilpotent,” meaning its operations become trivial after some steps. Essentially, Engel's theorem provides a way to recognize when complex algebraic systems have a certain kind of simplicity, where repeated interactions diminish to nothing, revealing their fundamental nature.