Engel's theorem
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. 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 . Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as . Statements 
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.) Proof 
We prove the following form of the theorem: 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–Chevalley decomposition#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. See also 
 Lie's theorem
 Heisenberg group

Notes

Citations

Works cited

Representation theory of Lie algebras
Theorems in representation theory