It is clear that any associative operator algebra can be seen as an operator Lie algebra. In this talk, we will talk about the relationships between operator Lie algebras and the associative algebras generated by them, and focus on two problems. There are Wojtynskii’s problem (Is the associative algebra generated by a Lie subalgebra of quasinilpotent operators also composed of quasinilpotent operators? ) and Shulman-Turovskii’s problem (Is a Banach algebra generated by an Engel Lie subalgebra an Engel algebra?)