From a physicist point of view I want to mention this trick and its generalization for operators:

      "Two commuting matrices are simultaneously diagonalizable"

(for physicists all matrices are diagonalizable). Of course the idea is that if you know the eigenvectors of one matrix/operator then diagonalizing the other one is much easier. Here are some applications.

1)The system is translation invariant : Because the eigenvectors of the translation operator are $e^{ik.x}$, then one should use the Fourier transform. It solves all the wave equations for light, acoustics, of free quantum electrons or the heat equation in homogeneous media.

2)The system has a discrete translation symmetry: The typical system is the atoms in a solid state that form a crystal. We have a discrete translation operator $T_a\phi(x)=\phi(x+a)$ with $a$ the size of the lattice and then we should try $\phi_k(x+a)=e^{ik.a}\phi_k(x)$ as it is an eigenvector of $T_a$. This gives the Bloch-Floquet theory where the spectrum is divided into band structure. It is one of the most famous model of condensed matter as it explains the different between conductors or insulators.

3)The system is rotational invariant: One should then use and diagonalize the rotation operator first. This will allow us to find the eigenvalue/eigenvectors of the Hydrogen atom. By the way we notice the eigenspace of the Hydrogen are stable by rotation and are therefore finite dimension representations of $SO(3)$. The irreducible representations of $SO(3)$ have dimension 1,3,5,... and they appears, considering also the spin of the electron, as the columns of the periodic table of the elements (2,6,10,14,...).

4)$SU(3)$ symmetry: Particle physics is extremely complicated. However physicists have discovered that there is an underlying $SU(3)$ symmetry. Then considering the representations of $SU(3)$ the zoology of particles seems much more organized (A, B).