Quaternions Defined by Matrices

Theorem
Let $$\mathbf 1, \mathbf i, \mathbf j, \mathbf k$$ denote the following four elements of the matrix space $$\mathcal M_\C \left({2}\right)$$:


 * $$\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:

$$ $$ $$ $$

Proof
This is demonstrated by straightforward application of conventional matrix multiplication:

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