  We describe subalgebras of the Lie algebra $\mf{gl}(n^2)$ 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 $A$.
