Norms of generalized derivations with values in symmetric spaces

Let $B(\mathcal H)$ be the $*$-algebra of all bounded linear operators on a Hilbert space $\mathcal H$. A classical result due to Stamplfi shows that, for any $a\in B(\mathcal H)$, the norm of the inner derivation $δ_a=[a,\cdot]$ on $B(\mathcal H)$ is given by $\|δ_a\|_{B(\mathcal H)\to B(\mathcal H