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}}