I just finished an exam where this was the final question. I could not for the life of me figure out how to do this. My first guess was to use the fact that there is a surjective homomorphism $I \otimes I \to I^2$ but this doesn't work since $I$ is proper and $I^2 \subset I$. My next idea was to work backward to find a homomorphism $\phi: A \to I \otimes I$ by determining $\phi(1)$. Expanding out $$(2a + (1 + \sqrt{-5})b) \otimes (2\alpha + (1 + \sqrt{-5})\beta)$$ did not yield a simple expression which suggested a simple generator. This is where my time essentially ran out so I resorted to trying $\phi(1) = 2 \otimes (1 + \sqrt{-5}) + (1 + \sqrt{-5}) \otimes 2$ but this idea didn't really pan out. Is there something big I'm missing here? I'm still having a lot of trouble seeing what I was supposed to do.
Thank you in advance for you help.