there is a statement in quantummechanics (physics) that commuting Hermitian operators in Hilbert space have simultaneous eigenstates. There is another statement that these operators are inherited from a common operator.
For example, let A and B Hermitian operators with $AB=BA$. Then, there exists an operator $X$ that $A=a(X)$ and $B=b(X)$, where $a,b$ are functions. How could you prove these two statements?