How to Explicitly Compute Charts for a Levelset Submanifold

While doing a computation with my friend Shane the other day, we realized we needed to explicitly compute a local chart near the identity of $SL_2(\mathbb{R})$. It took us longer than I’d like to admit to figure out how to do this (especially since it’s so geometrically obvious in hindsight), and so I want to write down the process for future grad students looking to just do a computation! If you…