Local characterization of Jordan $\ast$-derivations on $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional real Hilbert space, and $\mathcal{B}(H)$ the algebra of all bounded linear operators on $H$. Assume that $\delta:\mathcal{B}(H)\rightarrow\mathcal{B}(H)$ is a real linear map and $P\in\mathcal{B}(H)$ is zero, or the unit element, or a nontrivial idempotent with infinite-dimensional range and infinite-dimensional kernel. It is shown that $\delta$ satisfies $\delta(A^2)=\delta(A)A^*+A\delta(A)$ for all $A\in\mathcal{B}(H)$ with $A^2=P$ if and only if $\delta$ is an inner Jordan $\ast$-derivation. An example is also given to illustrate that this is not necessarily true when $H$ is finite-dimensional.