= Four-point regular representation of the Klein four-group
{title2=$P=(12)(34),\quad Q=(13)(24),\quad PQ=(14)(23)$}
The <permutation matrices> of the three displayed double transpositions, together with $I_4$, form a <faithful representation> of the <Klein four-group>. They satisfy $P^2=Q^2=I_4$ and $PQ=QP$. Every nonidentity element fixes no basis vector, so this is its four-dimensional <regular representation>.
The constant vector $(1,1,1,1)^T$ spans a nonzero proper <invariant subspace>, proving reducibility. More completely, the columns of
$$
S=\frac12\begin{pmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&1&-1\\1&-1&-1&1\end{pmatrix}
$$
form a common <orthonormal basis> of <eigenvectors>. Direct multiplication gives $S^TPS=\operatorname{diag}(1,1,-1,-1)$ and $S^TQS=\operatorname{diag}(1,-1,1,-1)$. Thus the representation splits into the four distinct real one-dimensional characters, each once. The basis change is a normalized <Hadamard matrix>.
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