I am going to get the principal angles between 2 subspaces in $\mathbb{R^4}$ according to this hint. The 2 matrices involved are $A = \begin{bmatrix} a & -b \\ b & a \\ c & -d \\ d & c \end{bmatrix}$ and $B = \begin{bmatrix} e & -f \\ f & e \\ g & -h \\ h & g \end{bmatrix}$ respectively. Evaluating $A^T B$ yields $\begin{bmatrix} ae + bf + cg + dh & -af + be - ch + dg \\ -be + af - dg + ch & ae + bf + cg + dh \end{bmatrix}$ which is already quite messy. So is there a quick way to find out $\sigma(A^T B)$, in particular the larger one, considering $A^T B$ is only a $2 \times 2$ matrix? Thanks in advance.
Edit: if I further simplify the notations, then $A^T B$ yields $\begin{bmatrix} \alpha & -\beta \\ \beta & \alpha \end{bmatrix}$ and I got the singular values to be $\sqrt{\alpha^2 + \beta^2}$ unless $\alpha = \beta = 0$. Would anyone help verify this?