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−1y−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.
- Shumyatsky, P.; Tortora, A.; Tota, A. (21 Feb 2014). "An Engel condition for orderable groups". arXiv: .
- In other words, n "["s and n copies of y, for example, [[[x,y],y],y], [[[[x,y],y],y],y]. [[[[[x,y],y],y],y],y], and so on.