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