It is well-known that any bounded normal operator $A$ can be written as $A= A_1+ i A_2$ where $A_1$ and $A_2$ are commuting bounded self-adjoint operators. This leads to a proof of spectral theorem of bounded normal operators using spectral theorem of bounded self-adjoint operators.
My question is, for an unbounded normal operator $A$, is there also such a decompostion $A=A_1+i A_2$ with $A_1$, $A_2$ (unbounded) self-adjoint? If so, is there an elementary proof of this without using spectral theorem of unbounded normal operators?