Theorem: Quaternions Defined by Matrices

Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\MM_\C} 2$:
:$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}
$
where $\C$ is the set of complex numbers.
Then $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are related to each other in the following way:
{{begin-eqn}}
{{eqn | l = \mathbf i \mathbf j = - \mathbf j \mathbf i
      | r = \mathbf k
}}
{{eqn | l = \mathbf j \mathbf k = - \mathbf k \mathbf j
      | r = \mathbf i
}}
{{eqn | l = \mathbf k \mathbf i = - \mathbf i \mathbf k
      | r = \mathbf j
}}
{{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k
      | r = -\mathbf 1
}}
{{end-eqn}}

Proof:
This is demonstrated by straightforward application of conventional matrix multiplication:
{{begin-eqn}}
{{eqn | l = \mathbf i \mathbf j
      | r = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} i \cdot 0 + 0 \cdot -1 & i \cdot 1 + 0 \cdot 0 \\ 0 \cdot 0 + -i \cdot -1 & 0 \cdot 1 + -i \cdot 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \mathbf k
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = -\mathbf j \mathbf i
      | r = -\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 0 \cdot i + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot -i \\ -1 \cdot i + 0 \cdot 0 & -1 \cdot 0 + 0 \cdot -i \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \mathbf k
      | c = 
}}
{{end-eqn}}
----
{{begin-eqn}}
{{eqn | l = \mathbf j \mathbf k
      | r = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 \cdot 0 + 1 \cdot i & 0 \cdot i + 1 \cdot 0 \\ -1 \cdot 0 + 0 \cdot i & -1 \cdot i + 0 \cdot 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
      | c = 
}}
{{eqn | r = \mathbf i
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = -\mathbf k \mathbf j
      | r = -\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 0 \cdot 0 + i \cdot -1 & 0 \cdot 1 + i \cdot 0 \\ i \cdot 0 + 0 \cdot -1 & i \cdot 1 + 0 \cdot 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
      | c = 
}}
{{eqn | r = \mathbf i
      | c = 
}}
{{end-eqn}}
----
{{begin-eqn}}
{{eqn | l = \mathbf k \mathbf i
      | r = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 \cdot i + i \cdot 0 & 0 \cdot 0 + i \cdot -i \\ i \cdot i + 0 \cdot 0 & i \cdot 0 + 0 \cdot -i \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \mathbf j
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = -\mathbf i \mathbf k
      | r = -\begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} i \cdot 0 + 0 \cdot i & i \cdot i + 0 \cdot 0 \\ 0 \cdot 0 + -i \cdot i & 0 \cdot i + -i \cdot 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \mathbf j
      | c = 
}}
{{end-eqn}}
----
{{begin-eqn}}
{{eqn | l = \mathbf i^2
      | r = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} i \cdot i + 0 \cdot 0 & i \cdot 0 + 0 \cdot -i \\ 0 \cdot i + -i \cdot 0 & -i \cdot 0 + -i \cdot -i \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\mathbf 1
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \mathbf j^2
      | r = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 \cdot 0 + 1 \cdot -1 & 0 \cdot 1 + 1 \cdot 0 \\ -1 \cdot 0 + 0 \cdot -1 & -1 \cdot 1 + 0 \cdot 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\mathbf 1
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \mathbf k^2
      | r = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} 0 \cdot 0 + i \cdot i & 0 \cdot i + i \cdot 0 \\ i \cdot 0 + 0 \cdot i & i \cdot i + 0 \cdot 0 \end{bmatrix}
      | c = 
}}
{{eqn | r = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
      | c = 
}}
{{eqn | r = -\mathbf 1
      | c = 
}}
{{end-eqn}}
----
{{begin-eqn}}
{{eqn | l = \mathbf i \mathbf j \mathbf k
      | r = \paren {\mathbf i \mathbf j} \mathbf k
      | c = Matrix Multiplication is Associative
}}
{{eqn | r = \mathbf k \mathbf k
      | c = from above: $\mathbf i \mathbf j = \mathbf k$
}}
{{eqn | r = -\mathbf 1
      | c = from above: $\mathbf k^2 = - \mathbf 1$
}}
{{end-eqn}}
{{qed}}