If $L^*$ is the formal self-adjoint of and $n$th order differential operator $L$ show that the formal adjoint of $L^*$ is $L$.

Question

If $L^*$ is the formal self-adjoint of and $n$th order differential operator $L$ show that the formal adjoint of $L^*$ is $L$.

Proof

By definition and hypothesis $L^* = L$. Therefore $\left(L^* \right)^* = \left(L\right)^* = L.$

Is it that easy or do I have to do some integration?