Ring Structures On $\mathbb {R} ^n$

Question

In the book of Musili it is written that $\mathbb{R}^n$ is a division ring under usual addition and multiplication for $n=1,2,4$. I have understood this. But after that he said, in those cases we cannot define any other multiplication in $\mathbb{R}^n$ to make it a division ring, i.e., the only multiplication '.' in $\mathbb{R}^n$ is the usual multiplication for which $(\mathbb{R}^n,+,.)$ is a division ring (where $n=1,2,4$). The proof is not given in the book. Can anyone help me?

Answer 1

I wouldn't say that $\Bbb R^n$ is a division algebra under "the usual multiplication," but rather using the multiplications that $\Bbb C$ and $\Bbb H$ induce.

The fact that the only finite dimensional associative $\Bbb R$ division algebras are the reals, the complexes, and the quaternions is precisely the Frobenius theorem.

It's too long to prove here, but you can search for a proof under that name.