I tried to write it in latex, but apparently it's too complicated for this website to read the code. Here is the link to it: https://ibb.co/dA4cV5
Anyway, it has been 2 years since I got it. Now, I don't remember what book I got it from. I'm 100% certain that I got it from a book! However, it's also possible that the book got the result from another original paper. Either way, I just want to look for a source that shows the derivation I have in the picture starting from "First, take $\Omega\in GL_m(\Bbb R)$..." This is a generalization of the Lorentz group. I know the J is a little bit different from the J in Lorentz group, but after you find $\theta$ in that case, with a little modification, you can find the $\theta$ in Lorentz group pretty easily.
I literally have been looking for the book and the paper for days but without any luck. Maybe with a few more pairs of eyes is what I need. Maybe some of you math experts will know what paper to look for right away when you see the derivation. Please help!
Some possible key searches include Lorentz group, Lorentz decomposition, Minkowski space, Lorentz boost...
I just found the book I've been looking for. It's call Special Relativity by Michael Tsamparlis. The proof starts on page 26 (section 1.7).