Matrix algebra question

Question

If $A,B$ are $n \times n$ complex matrices, Is it possible that $ABA-BAB =I$? I was able to solve a related problem " Can $AB-BA =I$ ? " using a trace argument, but that argument does not seem to extend to this problem

Answer 1

Putting $B=I$, you can solve the equation $A^2-A-I=0$ for $A$. Thus it is possible.

Answer 2

Consider the following matrices:

$$A = \left(\begin{array}{ccc} \frac{1 + \sqrt{5}}{2} & 0 \\ 0 & \frac{1 - \sqrt{5}}{2} \end{array}\right) \text{ and } B = \left(\begin{array}{ccc} \frac{-1 + \sqrt{5}}{2} & 0 \\ 0 & \frac{-1 - \sqrt{5}}{2} \end{array}\right)$$