= Solution
The explicit <matrices> are
$$
W=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad
X=\begin{pmatrix}i&0\\0&-i\end{pmatrix},\quad
Y=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\quad
Z=\begin{pmatrix}0&i\\i&0\end{pmatrix}.
$$
Direct <matrix multiplication> gives
$$
\boxed{WW=W,\quad XX=YY=ZZ=\widehat W,\quad XY=Z,\quad YZ=X,\quad ZX=Y.}
$$
The reverse products have the opposite signs: $YX=\widehat Z$, $ZY=\widehat X$, $XZ=\widehat Y$. Since $W$ is the <identity matrix> and $X^2=Y^2=Z^2=-W$,
$$
\boxed{W^{-1}=W,\qquad X^{-1}=\widehat X,\qquad Y^{-1}=\widehat Y,\qquad Z^{-1}=\widehat Z.}
$$
These are the usual quaternion multiplication relations in the <complex matrix representation of quaternions>.
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