Diagonalization & Matrix Powers · Challenge problem

Two matrices $A$ and $B$ commute and are simultaneously diagonalized by the same eigenvectors $(1,1)$ and $(1,-1)$: $A$ has eigenvalues $2$ and $5$ on these (respectively), and $B$ has eigenvalues $3$ and $-1$ on these same two eigenvectors. Since $A$ and $B$ share an eigenbasis, $AB$ has eigenvalues equal to the products of the corresponding eigenvalues of $A$ and $B$. Find $\det(AB - I)$.