I'm trying to solve the equation $$ -f(a+b,c,d)e^{a\cdot b}+f(a,b+c,d)e^{b\cdot c}-f(a,b,c+d)e^{c\cdot d}-f(a+d,b,c)e^{-a\cdot d}=0, $$ with $a,b\in \mathbb{R}^2$ and the scalar product $a\cdot b=a_1b_2-a_2b_1$. The function f is assumed to be Taylor expandable in the scalar procucts of its arguments. Moreover, since the vectors $a,b,c,d\in\mathbb{R}^2$, we have $$ (a\cdot b)(c\cdot d)+(b\cdot c)(a\cdot d)-(a\cdot c)(b\cdot d)=0. $$ This identity follows from the fact that in $\mathbb{R}^2$ no more than 2 vectors can be linearly independent.
So the question is simply how to solve this. I don't need a full answer, but it would be nice have a general direction to look into, cause I'm kind of in the dark what methods to use.
Thanks in advance!
-Edit: I'm trying to solve for the function $f$, given any vectors $a,b,c,d\in\mathbb{R}^2$. I've already tried Taylor expanding, hoping to get some set of differential equations and I'm currently trying to work out some power expansion of $f$ to see if I can find some recursive formula for the coefficients.