‎New elements in Hilbert algebras

In this paper‎, ‎the notion of commutator of elements of a Hilbert algebra are introduced and some properties are given‎. ‎The notions of involution element and Engel element in Hilbert algebras are introduced‎. ‎Many different characterizations of them are given‎. ‎Then‎, ‎left (right) $k$-Engel elements as a natural generalization of commutators are introduced‎, ‎and we discuss Engel elements‎, ‎which is defined by left and right commutators‎. ‎Finally‎, ‎we will also study relationships between these elements‎.