Problem: Let $A$ be a $n\times n$ matrix with real coefficients and let $k\geq 2$ be an integer. The fact that $A^k$ is diagonalizable in $\mathbb R$ does not imply that $A$ is. In fact, for example $$ A=\begin{pmatrix} 0&1\\0&0 \end{pmatrix}, $$ is not diagonalizable, but $A^2$ is the null matrix and is therefore trivially diagonalizable.
Question: Is there any non-trivial counterexample, i.e., matrices $A$ which are not diagonalizable and not nilpotent?