On Lie and associative algebras containing inner derivations

We describe subalgebras of the Lie algebra $\text{mf}gl(n^2)$ that contain all inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically closed field of characteristic 0). In a more general context where $A$ is a prime algebra satisfying certain technical restrictions, we establish a density theorem for the associative algebra generated by all inner derivations of $A$.