I am trying to understand a line from MacLachlan/Reid's The Arithmetic of Hyperbolic 3-Manifolds (it's in 3.4 if you have the book), that seems it should be elementary but I can't seem to find where it's coming from.

Suppose I have a vector space $V$ over a field $F$, and an alternating trilinear form $f:V^3\rightarrow F$. Suppose I also have a symmetric bilinear form $\sigma:V^2\rightarrow F$. Is it true in general that there must exist a constant $c\in F$ such that:

$f(x_1,x_2,x_3)f(y_1,y_2,y_3)=c$ det$\Big(\big(\sigma(x_i,y_j)\big)_{i,j}\Big)$?

I believe this is the property the authors quote to get their result, though it's possible I have overgeneralized it. The statement it's used for in the book is for the case when $V=M_2(\mathbb{C})$, $F=\mathbb{C}$, $f$ is $(x_1,x_2,x_3)\mapsto tr(x_1x_2x_3)$, and $\sigma$ is $(x_1,x_2)\mapsto tr(x_1x_2)$, with the added condition that we require the $x_i$ and $y_i$ to have trace $0$ (obviously products of these need not have trace zero), and in this case $c$ turns out to be $-\frac{1}{2}$. Obviously the equation in this case can be verified by a simple but tedious calculation, but this is not how the author derives it. He quotes the property above (or something close to it). If the statement is true, what is an elegant way to see it, perhaps from differential geometry, or just multilinear algebra? If I have overgeneralized the property, what is the correct one?