Engel's theorem

In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra ${\displaystyle {\mathfrak {g}}}$ is a nilpotent Lie algebra if and only if for each ${\displaystyle X\in {\mathfrak {g}}}$, the adjoint map

${\displaystyle \operatorname {ad} (X)\colon {\mathfrak {g}}\to {\mathfrak {g}},}$

given by ${\displaystyle \operatorname {ad} (X)(Y)=[X,Y]}$, is a nilpotent endomorphism on ${\displaystyle {\mathfrak {g}}}$; i.e., ${\displaystyle \operatorname {ad} (X)^{k}=0}$ 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).