Existence of solutions of matrix equation $A^4+I=O$

Question

Is there an $n \times n$ matrix $A$ over R, such that $A^4+I=O$ ?
I know that $det(A^4)=(det(A))^4=det(-I)=(-1)^n\geq 0$. Then $n$ has to be even, and $det(A)\neq 0$. Therefore, $A$ is invertible. But I have no idea what to do next.

Answer 1

If $n$ is even, consider a direct sum of $n/2$ copies of the $2\times2$ rotation matrix for an angle $\pi/4$.