On the product of two generalized derivations

Two elements $A$ and $B$ in a ring $\mathfrak{R}$ determine a generalized derivation $\delta _{A,B}$ on $\mathfrak{R}$ by setting $\delta _{A,B}(X)$ $=AX-XA$ for any $X$ in $\mathfrak{R}$. We characterize when the product $\delta _{C,D}\delta _{A,B}$ is a generalized derivation in the cases when the ring $\mathfrak{R}$ is the algebra of all bounded operators on a Banach space $\mathcal{E}$, and when $\mathfrak{R}$ is a $C^{*}$-algebra $\mathfrak{A}$. We use these characterizations to compute the commutant of the range of $\delta _{A,B}$.