product of a self-adjoint operator and a bounded operator, reference request

Question

Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$ and $B$ be a bounded self-adjoint operator from $B(H)$. Let $A=\int \lambda dE_\lambda$ be the spectral resolution. If $B$ commutes with every $E_\lambda$, then $AB$ is self-adjoint. I have seen that some paper use this fact and I can prove that but I think there should be some references for this fact. Does anyone know such a referece?