<s>[INST] Find answer of a system of equations Let $a,b,c\in \mathbb{R}$ that $a^{2}+b^{2}+c^{2}=1$. We want to find $x,y,z,w$ in the following equations:
$$\begin{cases}
\begin{align}
x^{2}+y^{2}&=\frac{1}{2}(1+a) \tag{1}\\
w^{2}+z^{2}&= \frac{1}{2}(1-a) \tag{2}\\
xw+yz&= \frac{1}{2} b \tag{3}\\
yw-xz&= \frac{1}{2} c.\tag{4}
\end{align}
\end{cases}$$
 [/INST] Hint (assuming $x,y,z,w$ are reals): let $u=y+ix$ and $v=w+iz$ then the system becomes:
$$
\begin{cases}
\begin{align}
|u|^2 &= \frac{1}{2}(1+a) \\
|v|^2 &= \frac{1}{2}(1-a) \\
uv & = \frac{1}{2}(c+ib)
\end{align}
\end{cases}
$$
From the first two equations $|u|^2|v|^2= \frac{1}{4}(1-a^2)=\frac{1}{4}(b^2+c^2)\,$. From the third equation $|uv|^2=\frac{1}{4}(c^2+b^2)\,$ as well, so the modulus part of the 3rd equation is redundant.
Then pick $u$ to be any complex number of modulus $\sqrt{\frac{1+a}{2}}\,$, and let $v=\frac{c+ib}{2u}$.
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