The CS decomposition is a way to write the singular value decomposition of a matrix with orthonormal columns. More specifically, taking the notation from these notes (pdf alert), consider a $(n_1+n_2)\times p$ matrix $Q$, with $$Q=\begin{bmatrix}Q_1 \\ Q_2\end{bmatrix},$$ where $Q_1$ has dimensions $n_1\times p$ and $Q_2$ has dimensions $n_2\times p$. Assume $Q$ has orthonormal columns, that is, $Q_1^\dagger Q_1+Q_2^\dagger Q_2=I$.
Then the CS decomposition essentially tells us that the SVDs of $Q_1$ and $Q_2$ are related. More specifically, there are unitaries $V, U_1, U_2$ such that \begin{aligned} U_1^\dagger Q_1 V=\operatorname{diag}(c_1,...,c_p), \\ U_2^\dagger Q_2 V=\operatorname{diag}(s_1,...,s_q), \end{aligned} with $c_i^2+s_i^2=1$ (from which the name of the decomposition comes). As far as I understand, this means that there is a set of orthonormal vectors $\{v_k\}_k$ such that both $\{Q_1 v_k\}_k$ and $\{Q_2 v_k\}$ are orthogonal sets of vectors (with some relations between their norms).
To prove that this is the case, I start by writing down the SVDs of $Q_1$ and $Q_2$, which tell us that there are unitaries $U_1, U_2, V_1, V_2$, and diagonal positive matrices $D_1, D_2$, such that \begin{aligned} Q_1= U_1 D_1 V_1^\dagger, \\ Q_2= U_2 D_2 V_2^\dagger. \end{aligned} The condition $Q_1^\dagger Q_1+Q_2^\dagger Q_2=I$ then translates into $$V_1 D_1^2 V_1^\dagger + V_2 D_2^2 V_2^\dagger=I.$$ Denoting with $v^{(i)}_k$ the $k$-th column of $V_i$, and $P^{(i)}_k\equiv v^{(i)}_k v^{(i)*}_k$ the associated projector, this condition can be seen to be equivalent to $$\sum_k (d^{(1)}_k)^2 P_k^{(1)}+\sum_k (d^{(2)}_k)^2 P_k^{(2)}=I,\tag A$$ where $d^{(i)}_k\equiv (D_i)_{kk}$.
Now, however, I'm a bit stuck into how to proceed from (A). It seems a generalisation of the things proved in this post and links therein, which show that if a sum of projectors gives the identity then the projectors must be orthogonal, but I'm not sure how to prove this in this case.