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In representation theory, a branch of mathematics, **Engel's theorem** states that a finite-dimensional Lie algebra is a nilpotent Lie algebra if and only if for each , the adjoint map

given by , is a nilpotent endomorphism on ; i.e., for some *k*.^{[1]} It is a consequence of the theorem, also called Engel's theorem, which says that if a Lie algebra of matrices consists of nilpotent matrices, then the matrices can all be simultaneously brought to a strictly upper triangular form. Note that if we merely have a Lie algebra of matrices which is nilpotent *as a Lie algebra*, then this conclusion does *not* follow (i.e. the naïve replacement in Lie's theorem of "solvable" with "nilpotent", and "upper triangular" with "strictly upper triangular", is false; this already fails for the one-dimensional Lie subalgebra of scalar matrices).

The theorem is named after the mathematician Friedrich Engel, who sketched a proof of it in a letter to Wilhelm Killing dated 20 July 1890 (Hawkins 2000, p. 176). Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as (Umlauf 2010).

Let be the Lie algebra of the endomorphisms of a finite-dimensional vector space *V* and a subalgebra. Then Engel's theorem states the following are equivalent:

- Each is a nilpotent endomorphism on
*V*. - There exists a flag such that ; i.e., the elements of are simultaneously strictly upper-triangulizable.

Note that no assumption on the underlying base field is required.

We note that Statement 2. for various and *V* is equivalent to the statement

- For each nonzero finite-dimensional vector space
*V*and a subalgebra , there exists a nonzero vector*v*in*V*such that for every

This is the form of the theorem proven in #Proof. (This statement is trivially equivalent to Statement 2 since it allows one to inductively construct a flag with the required property.)

In general, a Lie algebra is said to be nilpotent if the lower central series of it vanishes in a finite step; i.e., for = (*i*+1)-th power of , there is some *k* such that . Then Engel's theorem implies the following theorem (also called Engel's theorem): when has finite dimension,

- is nilpotent if and only if is nilpotent for each .

Indeed, if consists of nilpotent operators, then by 1. 2. applied to the algebra , there exists a flag such that . Since , this implies is nilpotent. (The converse follows straightforwardly from the definition.)

We prove the following form of the theorem:^{[2]} *if is a Lie subalgebra such that every is a nilpotent endomorphism and if *V* has positive dimension, then there exists a nonzero vector *v* in *V* such that for each *X* in .*

The proof is by induction on the dimension of and consists of a few steps. (Note the structure of the proof is very similar to that for Lie's theorem, which concerns a solvable algebra.) The basic case is trivial and we assume the dimension of is positive.

**Step 1**: Find an ideal of codimension one in .

- This is the most difficult step. Let be a maximal (proper) subalgebra of , which exists by finite-dimensionality. We claim it is an ideal of codimension one. For each , it is easy to check that (1) induces a linear endomorphism and (2) this induced map is nilpotent (in fact, is nilpotent as is nilpotent; see Jordan decomposition in Lie algebras). Thus, by inductive hypothesis applied to the Lie subalgebra of generated by , there exists a nonzero vector
*v*in such that for each . That is to say, if for some*Y*in but not in , then for every . But then the subspace spanned by and*Y*is a Lie subalgebra in which is an ideal of codimension one. Hence, by maximality, . This proves the claim.

**Step 2**: Let . Then stabilizes *W*; i.e., for each .

- Indeed, for in and in , we have: since is an ideal and so . Thus, is in
*W*.

**Step 3**: Finish up the proof by finding a nonzero vector that gets killed by .

- Write where
*L*is a one-dimensional vector subspace. Let*Y*be a nonzero vector in*L*and*v*a nonzero vector in*W*. Now, is a nilpotent endomorphism (by hypothesis) and so for some*k*. Then is a required vector as the vector lies in*W*by Step 2.

**^**Fulton & Harris 1991, Exercise 9.10..**^**Fulton & Harris 1991, Theorem 9.9..

- Erdmann, Karin; Wildon, Mark (2006).
*Introduction to Lie Algebras*(1st ed.). Springer. ISBN 1-84628-040-0. - Fulton, William; Harris, Joe (1991).
*Representation theory. A first course*. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. - Hawkins, Thomas (2000),
*Emergence of the theory of Lie groups*, Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98963-1, MR 1771134 - Hochschild, G. (1965).
*The Structure of Lie Groups*. Holden Day. - Humphreys, J. (1972).
*Introduction to Lie Algebras and Representation Theory*. Springer. - Umlauf, Karl Arthur (2010) [First published 1891],
*Über Die Zusammensetzung Der Endlichen Continuierlichen Transformationsgruppen, Insbesondre Der Gruppen Vom Range Null*, Inaugural-Dissertation, Leipzig (in German), Nabu Press, ISBN 978-1-141-58889-3