How to prove the Hamilton formula for quaternions

Question

$i^{2}=j^{2}=k^{2}=ijk=-1$

What kind of entities $i$, $j$ and $k$ are? and how to prove the aforementioned equation?

Answer 1

That equation is close to a definition of quaternions and therefore not much to prove.

The entities $i$, $j$, $k$ are different imaginary units. But if you refer to representations of the quaternions they can be different things depending on what representation you use. One well-known representation can be constructed using the Pauli matrices $$ i := -\imath \sigma_x = \begin{pmatrix}0&-\imath\\ -\imath&0\end{pmatrix}, \quad j := -\imath \sigma_x = \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad k := -\imath \sigma_x = \begin{pmatrix}-\imath&0\\0&\imath\end{pmatrix}, $$ where $\imath$ is the ordinary imaginary unit.