I want to find the cosine-sine (CS) decomposition of the following $4 \times 4$ unitary matrix:
$$ U = \begin{pmatrix} U_{11}&U_{12}\\ U_{21}& U_{22}\end{pmatrix},$$
where
$$U_{11} = \frac{1}{2}\begin{pmatrix} 1&-1\\ 1& 1\end{pmatrix}, \qquad U_{12} = \frac{1}{2}\begin{pmatrix} -1&1\\ -1& -1\end{pmatrix}$$
$$U_{21} = \frac{1}{2}\begin{pmatrix} 1&-1\\ 1& 1\end{pmatrix}, \qquad U_{22} = \frac{1}{2}\begin{pmatrix} 1&-1\\ 1& 1\end{pmatrix}$$
I tried finding the singular value decomposition of $U_{11}$ and $U_{22}$ with $\mbox{SVD} (U_{11}) = (L_1, D_1, R_1)$ and $\mbox{SVD} (U_{22}) = (L_2, D_2, R_2)$.
I am not sure what should be the next step in order to find the CS decomposition of matrix $U$? Any help would be appreciated.
The CSD is not as easy to compute and a stable algorithm is not that straightforward, see Paige, History and Generality of the CS Decomposition or Van Loan, Computing the CS and the generalized singular value decompositions. You can also google to find some freely available implementations, e.g. on GitHub.
The CSD will factorize the matrix $U$ with $U^T U=I$ as $$ U:= \begin{bmatrix} U_{11} & U_{12}\\ U_{21} & U_{22} \end{bmatrix}= \begin{bmatrix} Z_{1} & 0\\ 0 & Z_{2} \end{bmatrix} \begin{bmatrix} \Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} \begin{bmatrix} V_{1}^T & 0\\ 0 & V_{2}^T \end{bmatrix}, $$ where $1\succeq \Sigma_{ij}\succeq -1$ and $Z_i,V_i$ are unitary. In addition, $\Sigma_{11}^T\Sigma_{11}+\Sigma_{12}^T\Sigma_{12}=\Sigma_{21}^T\Sigma_{21}+\Sigma_{22}^T\Sigma_{22}=I$.
In your case, $U_{11}=U_{22}=U_{21}$ and $U_{12}=-U_{21}$, from which you immediately follow that the choice $Z_1=Z_2=:Z$, $V_1=V_2=:V$, $\Sigma_{11}=\Sigma_{21}=\Sigma_{22}=-\Sigma_{12}=:\Sigma$ should work.
You can set $V=I$, obtain $Z$ from normalizing the columns of $U_{11}$ and $\Sigma$ from the reciprocal of these factors, i.e.
$$ Z=\frac{1}{\sqrt{2}}\begin{bmatrix}1 & -1\\ 1 & 1 \end{bmatrix},\qquad \Sigma = \frac{\sqrt{2}}{2}\begin{bmatrix}1 & 0\\ 0 & 1 \end{bmatrix},\qquad V=\begin{bmatrix}1 & 0\\ 0 & 1 \end{bmatrix} $$ You will see that with this choice you have the properties of the factorization.