Find all $4 \times 4$ real matrices such that $A^3=I$.
The minimal polynomial must divide $x^3-1$. Since the matrix is real, the minimal polynomial must be either $x-1$ or $x^3-1$ (i.e., if it contains one of the complex roots of unity, it must contain the conjugate). Of course, we already know the identity matrix satisfies this equation. If $A$ is any other non-identity real matrix satisfying the above properties, then its characteristic polynomial must be $(x-1)(x^3-1)$, by similar reasoning above. This is as far as I could get.
edit: I missed a few cases, as pointed out in the comments. The minimal polynomial can also be $x^2+x+1$ which yields additional possible characteristic polynomials.