If we have the parallel that \(AJ\) is to \(A\) what \(-a\) is to \(a\), then what of the parallell \(ia\) to \(a\), consider the operation \[A \to J((JA)J) = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\bigg(\bigg(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix}\bigg)\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\bigg) =\begin{bmatrix} b & a \\ d & c\end{bmatrix} Nope just ends up at $-1$ equivalence\]