I have found the following Fibonacci Identity (and proved it).

If $F_n$ denotes the nth Fibonacci Number, we have the following identity \begin{equation} F_{n-r+h}F_{n+k+g+1} - F_{n-r+g}F_{n+k+h+1} = (-1)^{n+r+h+1} F_{g-h}F_{k+r+1} \end{equation} where $F_1 = F_2 = 1$, $r \leq n$, $h \leq g$, and $n, g, k \in \mathbb{N}$.

It is not too hard to show that this identity subsumes Cassini's Identity, Catalan's Identity, Vajda's Idenity, and d'Ocagne's identity to name a few.

I have done a pretty thorough literature review, and I have not found anything like this, but I am still wondering if anyone has seen this identity before? I found this by accident after noticing some patterns in some analysis work I was doing, so if this is already known I would be curious to see what the connections are. Thanks for your patience and input!