Let $A$ be a real $4\times4$ matrix such that $A^2+A+I = 0$. Can $A$ be orthogonal?

Hint: a real $2 \times 2$ matrix representation of $-\frac{1}{2} + \frac{\sqrt{3}}{2} i$ would be $$ \begin{bmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{bmatrix}. $$ Check that this matrix satisfies $A^2 + A + I = 0$ and is orthogonal. Now, can you think of a way to use this to create a $4 \times 4$ example?

You need only find a $2 \times 2$ example, say $B$ and then create $A= \operatorname{diag}(B,B)$.

What are the eigenvalues of $\begin{bmatrix} -\cos t & -\sin t \\ \sin t & -\cos t \end{bmatrix}$?