= Equivalent sheared representations of the Klein four-group
Let the <Klein four-group> have generators $X,Y$ and $Z=XY$, with $X^2=Y^2=I$ and $XY=YX$. For every real $p$, the matrices
$$
D(X)=-I_2,\qquad D(Y)=\begin{pmatrix}-1&-p\\0&1\end{pmatrix},\qquad D(Z)=-D(Y)
$$
define a <group representation>. Indeed, $D(Y)^2=I_2$, and $D(X)$ commutes with every matrix, so the group relations hold.
Set $S=\begin{pmatrix}1&-p/2\\0&1\end{pmatrix}$. A direct <change of basis> gives
$$
S^{-1}D(Y)S=\operatorname{diag}(-1,1),\qquad S^{-1}D(Z)S=\operatorname{diag}(1,-1).
$$
Consequently all these <representations> are <isomorphic representations>, with <character> values $(2,-2,0,0)$ on $(I,X,Y,Z)$. Each is a <direct sum> of the two <one-dimensional representations> whose values on $X$ are $-1$ and whose values on $Y$ are $+1$ and $-1$. A <group representation> need not be <orthogonal>; nonzero $p$ changes the basis rather than the represented group.
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