Let us consider a commutator of two diagonizable $n\times n$, i.e. square matrices, $A$ and $B$. The eigenvalues of $A$ are $\lambda^1,\lambda^2,...,\lambda^{n}$ The eigen values of $B$ are $\mu^1,\mu^2,...,\mu^{n}$ The (normalized) eigenvectors of $A$ are $\bf{a}^1,\bf{a}^2,...$ and of $B$ are $\bf{b}^1,\bf{b}^2,...$ We assume these eigenvalues and eigenvectors are all known, and eigenvalues of matrices may or may not have degenerecies.
The commutator we use here is $$\sigma_{ab}=[A,B] =AB-BA$$ We will lable the eigenvalues of the commutator as $\sigma_{ab}^1,\sigma_{ab}^2,...$ and the eigenvectors as $\bf\sigma_{ab}$
We want a method to calculate the eigenvalues and and associated eigenvectors of $\sigma_{ab}$ by computing directly from our known eigenvalues and eigenvectors of $A$ and $B$ (to be clear not by using $A$ and $B$ to compute $\sigma_{ab}$ and then its eigenvalues and eigenvectors)