Engel group

In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element, named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator [...x,y],y], ..., y] with n copies of y is trivial (where [x, y] means xyx&minus;1y&minus;1 or the [[Lie bracket). It is called an Engel element if it satisfies the Engel condition that it is n-Engel for some n.

A Lie group or Lie algebra is said to satisfy the Engel or n-Engel conditions if every element does. Such groups or algebras are called Engel groups, n-Engel groups, Engel algebras,  and n-Engel algebras.

Every nilpotent group or Lie algebra is Engel. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. gave examples of non-nilpotent Engel groups and algebras.