Prove that no 2 orthogonal matrices satisfy this equation $A^2-B^2=AB$ This question came up in my linear algebra finals, and I couldn't prove it. Could anyone help me?

Question: Prove that no $2$ orthogonal matrices satisfy this condition:  $A^2-B^2=AB$

Attempt: Assume $A$ and $B$ are orthogonal. Mutiplying $A^T, B^T$ to both sides, we have:
$$
A-A^TB^2=B \\
AB^T-A^TB=I
$$
Taking transpose of both sides:
$$
(AB^T-A^TB)^T=I\\
BA^T-B^TA=I
$$
Again multiplying $B, A$ to both sides:
$$
B^2A^T-A=B \\
B^2-A^2=BA
$$
So combining, I have $AB=-BA$, but after this I'm stuck. I thought of the fact that $2x2$ orthogonal matrices represent a rotation in the $\mathbb{R}^2$ space, and so they commute, but in general rotations do not commute with reflections and an $nxn$ orthogonal matrices do not commute. 
Hints appreciated.

Consider the traces of both sides of $AB^T-A^TB=I$.
