In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is a nilpotent Lie algebra if and only if for each X ∈ g {\displaystyle X\in {\mathfrak {g}}}, the adjoint map ad ⁡ : g → g, {\displaystyle \operatorname {ad} \colon {\mathfrak {g}}\to {\mathfrak {g}},} given by ad ⁡ = [ X, Y ] {\displaystyle \operatorname {ad} =[X,Y]}, is a nilpotent endomorphism on g {\displaystyle {\mathfrak {g}}} ; i.e., ad ⁡ k = 0 {\displaystyle \operatorname {ad} ^{k}=0} for some k. Wikipedia