for any lie algebra let and inductively define and an easy induction shows that . thus if is nilpotent, then is solvable.
conversely, suppose that is solvable. then, by Lie's theorem, has a basis such that the matrix of with respect to is upper triangular for every it is easy to see that if and are two upper triangular matrices, then is strictly upper triangular and hence nilpotent. thus is nilpotent for all so we have proved that is nilpotent for all and hence, by Engel's theorem, is nilpotent.