I am trying to solve the following exercise:
Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra.
I am aware of Frobenius' theorem that there are only three finite dimensional real associative division algebras. But the exercise would be pointless if the theorem was assumed.
Let $x,y \in \mathbb R^3$. Since $x y$ has to extend the complex multiplication:
$$xy = (x_1y_1-x_2y_2, x_1y_2 + x_2 y_1, ?)$$
I have no idea how I can proceed from here. Could someone tell me how to do this? It should be easy as the other exercises I did so far were also easy.