I came across a question:
Prove or disprove: $A$ is a square matrix of size $n$ over $\mathbb R$, $P_{(\lambda)}=\lambda^k-1,P_{(A)}=0, k\geq 1 \implies A$ is diagonalizable over $\mathbb R$
So the obvious example is $A=I$ where you get that $I^3=I$ but i'm not sure if there are any other matrices over $\mathbb R$ can be powered into $I$.