Let $F,G:\mathbb{C}^2\rightarrow\mathbb{H}\otimes\mathbb{R}^4$ be two functions with components $F_0,F_1,F_2,F_{1,2},G_0,G_1,G_2,G_{1,2}:\mathbb{C}^2\rightarrow\mathbb{H}$ with quaternionic values. Assume that for $K\in\{0,1,2,\{1,2\}\}$
$$F_K\left(\overline{z}_1,z_2\right)=\left\{\begin{array}{l}
F_K(z_1,z_2)\qquad \text{if }\ 1\notin K\\
-F_K(z_1,z_2)\qquad \text{if }\ 1\in K
\end{array}\right.$$
and
$$F_K\left(z_1,\overline{z}_2\right)=\left\{\begin{array}{l}
F_K(z_1,z_2)\qquad \text{if }\ 2\notin K\\
-F_K(z_1,z_2)\qquad \text{if }\ 2\in K.
\end{array}\right.$$
The same holds for $G_K$. These functions are usually called Stem functions.
Is there any hope to deduce some property of the components of $F$ and $G$ by the fact that they solve the following equation
$$F_{12}G_0+F_1G_2+F_2G_1+F_0G_{12}=0?$$
I'm looking for something like $F_{1,2}=G_{1,2}=0$, but this is not the case in general.
P.S. What about if we suppose $F$ and $G$ holomorphic, adding these "Cauchy-Riemann equations" of the form
$$\dfrac{\partial F_0}{\partial\alpha_1}=\dfrac{\partial F_1}{\partial\beta_1},\quad\dfrac{\partial F_0}{\partial\beta_1}=-\dfrac{\partial F_1}{\partial\alpha_1},\quad\dfrac{\partial F_2}{\partial\alpha_1}=\dfrac{\partial F_{1,2}}{\partial\beta_1},\quad\dfrac{\partial F_2}{\partial\beta_1}=-\dfrac{\partial F_{1,2}}{\partial\alpha_1},$$
$$\dfrac{\partial F_0}{\partial\alpha_2}=\dfrac{\partial F_2}{\partial\beta_2},\quad\dfrac{\partial F_0}{\partial\beta_2}=-\dfrac{\partial F_2}{\partial\alpha_2},\quad\dfrac{\partial F_1}{\partial\alpha_2}=\dfrac{\partial F_{1,2}}{\partial\beta_2},\quad\dfrac{\partial F_1}{\partial\beta_2}=-\dfrac{\partial F_{1,2}}{\partial\alpha_2},$$
which hold for $G$ as well?
Thank you to anyone for the help.