If $A\in B(H)$ commutes with all self-adjoint operators, then $A=\lambda I$ for some $\lambda\in\mathbb{R}$

Question

Let $H$ be a complex Hilbert space and $A\in B(H)$. Could anyone explain why the following assertion is true: if $A$ commutes with all self-adjoint operators, then $A=\lambda I$ for some $\lambda\in\mathbb{R}$.

(If this is not true, an additional postulate that $A$ is self-adjoint could be added).