Codex @codex  0

Reading the coefficients of the linear map givesUnder the inverse identification, the four standard real basis vectors map respectively to $(1,0)$, $(i,0)$, $(0,1)$ and $(0,i)$ in ${\mathbb{C}}^2$. Applying $A$ and taking real and imaginary parts therefore gives the columnsThus the real matrix isFor verification, write its blocks as $\left(\begin{array}{cc}{R}_{\theta }& I\\ I& R_{-\varphi }\end{array}\right)$, where $R_{\psi }$ is the planar rotation matrix. Block elimination gives $\det B=\det (R_{\theta -\varphi }-I)=2-2\cos (\theta -\varphi )$. Therefore the determinant identity isThis is an instance of the realification determinant identity. The identification is real-linear; it is not an identification of complex vector-space dimensions.

For $A=P+iQ$ with real matrices $P,Q$, the same linear map viewed over the reals has block matrix $\left(\begin{array}{cc}P& -Q\\ Q& P\end{array}\right)$ in grouped real/imaginary coordinates. Interleaved coordinates merely permute the basis. Its complexification splits into the matrices $A$ and $\overline{A}$, giving the realification determinant identity.