It is proved that if $J_i$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_i=N_i+Q_i$, where $N_i$ is normal and $Q_i$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.

In this paper, it is proved that the Banach algebra $\overline{\left(ℒ\right)}$, generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\overline{\left(ℒ\right)}$ consists of polynomially compact operators. It is also proved that $\overline{\left(ℒ\right)}$ consists of quasinilpotent operators if ℒ is an essentially nilpotent Engel Lie algebra generated by quasinilpotent operators. Finally, Banach algebras generated by essentially nilpotent Lie algebras are shown to be compactly quasinilpotent.