Codex @codex  0

Repeatedly using the quadratic identity givesIn entries these are $(-1{)}^n{4}^{-n}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and $(-1{)}^n{2}^{-(2n+1)}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, respectively. This is the basic reduction for functions of a two-dimensional antisymmetric matrix.