How is it possible to show that the norm on the elements of Hamilton Quaternions is such that $N(\alpha \beta) = N(\alpha)N(\beta)$?

Question

Let $\alpha,\beta \in \mathbb{H}$ and the norm on $\mathbb{H}$ is defined as $N(\alpha) = \alpha \bar{\alpha}$.

How is it possible to show that the norm on the elements of Hamilton Quaternions is such that $N(\alpha \beta) = N(\alpha)N(\beta)$?

I know that $\bar{\alpha} \bar{\beta} = \bar{\alpha \beta}$, but I don't know how to use it?

Answer 1

The problem is that, rather, we have the identity $$\overline{\alpha\beta}=\bar\beta\bar\alpha$$ for any $\alpha,\beta\in\Bbb H.$ Hence, $$N(\alpha\beta)=\alpha\beta\bar\beta\bar\alpha=\alpha N(\beta)\bar\alpha.$$

All that remains is to justify that $N(\beta)$ commutes multiplicatively with every element of $\Bbb H,$ regardless of our choice of $\beta\in\Bbb H.$