A Theorem on Engel Groups

For any pair of elements x, y of a group, we write
[x,y]for the elements x^-1 y^-1 xy,
[x₁,...,x_k] for [[x₁,...,x_k-1],x_k] where k>0,
[(x,y)_k] for [x,y,...,y] where k>0.