Starting with these four matrices:

$\displaystyle I = {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right) }, \quad \quad \sigma_z = {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right) }, \quad \quad \sigma_x = {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right) } , \quad \quad {\boldsymbol{\hat{\jmath}}} = {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right) }$
Where:

Multiplying by z:

$\displaystyle \begin{array}{cccc} {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} \end{array}$

Multiplying by x:

$\displaystyle \begin{array}{cccc} {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} \end{array}$

Multiplying by i:

$\displaystyle \begin{array}{cccc} {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\boldsymbol{\hat{\imath}}} = {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\boldsymbol{\hat{k}}} = {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & -\sigma_y = {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} \end{array}$

This gives us 16 matrices with the following multiplication table.

Pauli matrices are in blue and quaternions are in green.
 $\displaystyle \begin{array}{cccc|cccc|cccc|cccc} {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} \\ {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} \\ {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} -1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{green} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} \\ {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} \\ \hline {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -x & 0 \\ 0 & -x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} -1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ -z & 0 \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -i & 0 \\ 0 &-i \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -z & 0 \\ 0 & \phantom{-}z \end{array} \right)} \\ {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -z & 0 \\ 0 & -z \end{array} \right)} \\ \hline {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{green} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} -1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} \\ \hline {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -i & 0 \\ 0 & -i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} -1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} \\ {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right)} \\ {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ -x & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -1 & 0 \\ 0 &-1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} -1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} \\ {\color{red} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & \phantom{-}i \\ \phantom{-}i & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{green} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} -1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right)} \end{array}$

All 16 matrices can be formed by multiplication from just 5 matrices:

$\displaystyle \begin{array}{c|cccc} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} \\ \hline {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -1 \\ \phantom{-}1 & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} -i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} \\ {\color{blue} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & -1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}1 \\ -1 & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} \\ {\color{blue} \left( \begin{array}{rr} 0 & \phantom{-}i \\ -i & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & -i \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ -z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -x \\ \phantom{-}x & 0 \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & \phantom{-}x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ \phantom{-}x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}x & 0 \\ 0 & -x \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & -z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} \\ {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & \phantom{-}z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}z \\ \phantom{-}z & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}z & 0 \\ 0 & -z \end{array} \right)} & {\color{black} \left( \begin{array}{rr} 0 & \phantom{-}x \\ -x & 0 \end{array} \right)} & {\color{black} \left( \begin{array}{rr} \phantom{-}i & 0 \\ 0 & \phantom{-}i \end{array} \right)} & {\color{red} \left( \begin{array}{rr} \phantom{-}1 & 0 \\ 0 & \phantom{-}1 \end{array} \right)} \end{array}$