Let B(X) denote the algebra of operators on a complex Banach space X, H(X) = {h ${in}$ B(X) : h is hermitian}, and J(X) = {x ${in}$ B(X) : x = $x_1$ + $ix_2$, $x_1$ and $x_2$ ${in}$ H(X)}. Let ${delta}_a$ ${in}$ B(B(X)) denote the derivation ${delta}_a$ = ax - xa. If J(X) is an algebra and ${delta}_a^{-1}(0){subseteq}{delta}_{a^*}^{-1}(0)$ for some $a{in}J(X)$, then ${parallel}a{parallel}{leq}{parallel}a-(x^*x-xx^*){parallel}$ for all $x{in}J(X){cap}{delta}_a^{-1}(0)$. The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = $C_p$, the von Neumann-Schatten p-class, are considered.