= Quarter-turn action on diagonal quadratic forms
{title2=$D(J)=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad\ker D=\{I,-I\}$}
Let $J$ be the plane <rotation matrix> through $\pi/2$, generating a <cyclic group> of order four. A <quadratic form> $ax^2+by^2$ has its two coefficients exchanged by a quarter-turn and unchanged by a half-turn. Thus the pullback action on this two-dimensional space gives the displayed <group representation>: $D(J^k)=D(J)^k$, with $D(J)^2=I$. It is not a <faithful representation>, because its <kernel of a group homomorphism> is $\{I,J^2\}$. Its two invariant coefficient lines correspond to $x^2+y^2$ and $x^2-y^2$, on which a quarter-turn acts by $+1$ and $-1$, respectively. Restricting to this diagonal space is valid for these signed coordinate permutations; a general <linear transformation> can introduce an $xy$ term.
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