Engel’s theorem

Before proceeding, it will be useful to recall the definition of a nilpotent Lie algebraMathworldPlanetmath. Let 𝔀 be a Lie algebraMathworldPlanetmath. The lower central series of 𝔀 is defined to be the filtrationMathworldPlanetmathPlanetmath of ideals




To say that 𝔀 is nilpotentPlanetmathPlanetmathPlanetmath is to say that the lower central series has a trivial termination, i.e. that there exists a k such that


or equivalently, that k nested bracket operationsMathworldPlanetmath always vanish.

Theorem 1 (Engel)

Let gβŠ‚EndV be a Lie algebra of endomorphismsPlanetmathPlanetmathPlanetmath of a finite-dimensionalPlanetmathPlanetmath vector spaceMathworldPlanetmath V. Suppose that all elements of g are nilpotent transformations. Then, g is a nilpotent Lie algebra.

Lemma 1

Let X:V→V be a nilpotent endomorphism of a vector space V. Then, the adjoint action


is also a nilpotent endomorphism.


Suppose that


for some kβˆˆβ„•. We will show that


Note that




are the endomorphisms corresponding, respectively, to left and right multiplicationPlanetmathPlanetmath by X. These two endomorphisms commute, and hence we can use the binomial formula to write


Each of terms in the above sum vanishes because



Lemma 2

Let g be as in the theoremMathworldPlanetmath, and suppose, in addition, that g is a nilpotent Lie algebra. Then the joint kernel,


is non-trivial.


We proceed by inductionMathworldPlanetmath on the dimension of 𝔀. The claim is true for dimension 1, because then 𝔀 is generated by a single nilpotent transformation, and all nilpotent transformations are singularPlanetmathPlanetmath.

Suppose then that the claim is true for all Lie algebras of dimension less than n=dim⁑𝔀. We note that π’Ÿ1⁒𝔀 fits the hypotheses of the lemma, and has dimension less than n, because 𝔀 is nilpotent. Hence, by the induction hypothesis


is non-trivial. Now, if we restrict all actions to V0, we obtain a representation of 𝔀 by abelianMathworldPlanetmath transformations. This is because for all a,bβˆˆπ”€ and v∈V0 we have


Now a finite number of mutually commuting linear endomorphisms admits a mutual eigenspaceMathworldPlanetmath decomposition. In particular, if all of the commuting endomorphisms are singular, their joint kernel will be non-trivial. We apply this result to a basis of 𝔀/π’Ÿ1⁒𝔀 acting on V0, and the desired conclusionMathworldPlanetmath follows. QED

Proof of the theorem.

We proceed by induction on the dimension of 𝔀. The theorem is true in dimension 1, because in that circumstance π’Ÿ1⁒𝔀 is trivial.

Next, suppose that the theorem holds for all Lie algebras of dimension less than n=dim⁑𝔀. Let π”₯βŠ‚π”€ be a properly contained subalgebraMathworldPlanetmathPlanetmath of minimum codimension. We claim that there exists an aβˆˆπ”€ but not in π”₯ such that [a,π”₯]βŠ‚π”₯.

By the induction hypothesis, π”₯ is nilpotent. To prove the claim consider the isotropy representation of π”₯ on 𝔀/π”₯. By Lemma 1, the action of each a∈π”₯ on 𝔀/π”₯ is a nilpotent endomorphism. Hence, we can apply Lemma 2 to deduce that the joint kernel of all these actions is non-trivial, i.e. there exists a aβˆˆπ”€ but not in π”₯ such that


for all b∈π”₯. Equivalently, [π”₯,a]βŠ‚π”₯ and the claim is proved.

Evidently then, the span of a and π”₯ is a subalgebra of 𝔀. Since π”₯ has minimum codimension, we infer that π”₯ and a span all of 𝔀, and that

π’Ÿ1β’π”€βŠ‚π”₯. (1)

Next, we claim that all the π’Ÿk⁒π”₯ are ideals of 𝔀. It is enough to show that


We argue by induction on k. Suppose the claim is true for some k. Let b∈π”₯,cβˆˆπ’Ÿk⁒π”₯ be given. By the Jacobi identityMathworldPlanetmath


The first term on the right hand-side in π’Ÿk+1⁒π”₯ because [a,b]∈π”₯. The second term is in π’Ÿk+1⁒π”₯ by the induction hypothesis. In this way the claim is established.

Now a is nilpotent, and hence by Lemma 1,

ad(a)n=0 (2)

for some nβˆˆβ„•. We now claim that


By (1) it suffices to show that

[𝔀,[…[π”€βžn⁒ times,π”₯]…]]βŠ‚π’Ÿ1π”₯.



this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to

[𝔀1,[…[𝔀1⏞n⁒ times,π”₯1]…]]=0.

However, π”₯1 is abelian, and hence, the above follows directly from (2).

Adapting this argument in the obvious fashion we can show that


Since π”₯ is nilpotent, 𝔀 must be nilpotent as well. QED

Historical remark.

In the traditional formulation of Engel’s theorem, the hypotheses are the same, but the conclusion is that there exists a basis B of V, such that all elements of 𝔀 are represented by nilpotent matricesMathworldPlanetmath relative to B.

Let us put this another way. The vector space of nilpotent matrices Nil, is a nilpotent Lie algebra, and indeed all subalgebras of Nil are nilpotent Lie algebras. Engel’s theorem asserts that the converseMathworldPlanetmath holds, i.e. if all elements of a Lie algebra 𝔀 are nilpotent transformations, then 𝔀 is isomorphic to a subalgebra of Nil.

The classical result follows straightforwardly from our version of the Theorem and from Lemma 2. Indeed, let V1 be the joint kernel 𝔀. We then let U2 be the joint kernel of 𝔀 acting on V/V0, and let V2βŠ‚V be the subspacePlanetmathPlanetmath obtained by pulling U2⁒x back to V. We do this a finite number of times and obtain a flag of subspaces


such that


for all k. The choose an adapted basis relative to this flag, and we’re done.

Title Engel’s theorem
Canonical name EngelsTheorem
Date of creation 2013-03-22 12:42:25
Last modified on 2013-03-22 12:42:25
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Theorem
Classification msc 17B30
Classification msc 15A57