Let $\mathcal{A}: \mathbb{C}^{2} \rightarrow \mathbb{C}^{2}$ be the linear map

$$\mathcal{A}\left(\begin{array}{c} z \\ w \end{array}\right)=\left(\begin{array}{c} z \mathrm{e}^{i \theta}+w \\ w \mathrm{e}^{-i \phi}+z \end{array}\right)$$

where $\theta$ and $\phi$ are real constants. Write down the matrix $A$ of $\mathcal{A}$ with respect to the standard basis of $\mathbb{C}^{2}$ and show that $\operatorname{det} A=2 i \sin \frac{1}{2}(\theta-\phi) \exp \left(\frac{1}{2} i(\theta-\phi)\right)$.

Let $\mathcal{R}: \mathbb{C}^{2} \rightarrow \mathbb{R}^{4}$ be the invertible map

$$\mathcal{R}\left(\begin{array}{c} z \\ w \end{array}\right)=\left(\begin{array}{l} \operatorname{Re} z \\ \operatorname{Im} z \\ \operatorname{Re} w \\ \operatorname{Im} w \end{array}\right)$$

and define a linear map $\mathcal{B}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}$ by $\mathcal{B}=\mathcal{R} \mathcal{A} \mathcal{R}^{-1}$. Find the image of each of the standard basis vectors of $\mathbb{R}^{4}$ under both $\mathcal{R}^{-1}$ and $\mathcal{B}$. Hence, or otherwise, find the matrix $B$ of $\mathcal{B}$ with respect to the standard basis of $\mathbb{R}^{4}$ and verify that $\operatorname{det} B=|\operatorname{det} A|^{2}$.