Let $x,y,z,t\in [0,1]$ such that $x^2+y^2=z^2+t^2=1$ and $\alpha,\omega \in [0,\frac{\pi}{2}]$ be given parameters. I'm trying to find out when there is an orthogonal (real) $4\times 4$ matrix $$P = \begin{pmatrix}p_{11} & p_{12} & p_{13} & p_{14} \\ p_{21} & p_{22} & p_{23} & p_{24} \\ p_{31} & p_{32} & p_{33} & p_{34} \\ p_{41} & p_{42} & p_{43} & p_{44} \\ \end{pmatrix}$$ such that $$P \begin{pmatrix}xe^{i\alpha} \\0 \\y \\0 \end{pmatrix}= \begin{pmatrix}ze^{i\omega}c_1 \\ze^{i\omega}c_2 \\tc_1 \\tc_2 \end{pmatrix}=\begin{pmatrix}ze^{i\omega} \\t \end{pmatrix} \otimes \begin{pmatrix}c_1 \\c_2 \end{pmatrix}$$ for some arbitrary complex numbers $c_1$ and $c_2$. i.e. Given the parameters $(x,y,\alpha)$ and $(z,t,\omega)$ how can one check that there exists a real orthogonal matrix $P$ that does the job? Is there a way to find constraints on the parameters?


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