EN
In this paper, it is proved that the Banach algebra $\overline{𝓐(ℒ)}$, generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\overline{𝓐(ℒ)}$ consists of polynomially compact operators. It is also proved that $\overline{𝓐(ℒ)}$ 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.