I was reading Griffiths, and he made a statement that if two operators commute with the Hamiltonian, but do not commute with each other, then the energy spectrum has to be degenerate. He gave the following reasoning:
If there is not a complete set of simultaneous eigenstates of all three operators, does that mean that there is some state in the Hilbert space that cannot be written as a linear combination of the simultaneous eigenstates? Why do we know from that that "there must be some $|\psi\rangle$ such that $\Lambda|\psi\rangle$ is distinct from $|\psi\rangle$?" (the underlined sentence).