Eigenvalues of product/sum of two matrices
$A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has eigenvalue 1, $B=\begin{pmatrix}0&2\\2&0\end{pmatrix}$ has eigenvalue -2
$A+B=\begin{pmatrix}0&3\\3&0\end{pmatrix}$ does not have eigenvalue $1-2=-1$
$AB=\begin{pmatrix}2&0\\0&2\end{pmatrix}$ does not have eigenvalue $1\cdot-2=-2$
Ok trying again. Take $$A = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0\end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 0 & 1\\ 0 & 2 & 0\\ 1 & 0 & 1\end{bmatrix}\,.$$
These matrices commute, neither is diagonal, and neither is triangular.
Eigenvalues of $A$: $-1, 1, 0$.
Eigenvalues of $B$: $2, 2, 0$.
Eigenvalues of $A+B$: $3,2,-1$.
Eigenvalues of $AB$: $2,0,0$.
So take $\lambda = -1$ and $\mu = 2$.