Reading "Mathematical Physics: Classical Mechanics" by A. Knauf, I found the following statement:
The positive symmetric matrices with determinant 1 can be written as $$ \begin{vmatrix} A & B \\ B & C \\ \end{vmatrix} $$ with $A>0$, $B \in \mathbb R$, $C=\frac{1+B^2}{A}$.
I'm on self-teaching, so even if it's maybe a well-known property, I've never encountered it.
From what "process" stems this equality, and is it valid for any "positive symmetric matrix with determinant 1", or am I missing some context?
By Sylvester's criterion, a real symmetric $2\times2$ matrix $M=\pmatrix{A&B\\ B&C}$ is positive definite if and only if $A>0$ and $\det M=AC-B^2>0$. So, if you want $\det M=1$, you need $C=\frac{1+B^2}A$.