Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if $$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and $B$ have the same characteristic polynomial, thus the same eigenvalues.

I'm interested in pairs of matrices $A$ and $B$ that satisfy all those equations except the last one, i.e. $$\det(A)=\det(B)$$ $$\mathrm{tr}(A)=\mathrm{tr}(B)$$ $$\mathrm{tr}(A^2)=\mathrm{tr}(B^2)$$ but $\mathrm{tr}(A^3) \neq \mathrm{tr}(B^3)$.

Does anyone know how to generate such matrices? Have they ever been studied? A reference would be nice.