Wow - that's a big one!
Root-wise, you'll want to get as much as possible. Digging looks like it could be tough as I see a sawed-off trunk that may have caused the sidewalk to buckle.
Foliage wise, it's also a good idea to keep as much as possible. Some things that come to mind:
- It looks like some foliage will need to be cut to simply make space for digging
- If you find you get no roots on one side, feel free to reduce the foliage there
- If there are any obvious unnecessary branches (long, straight) those are good candidates for cutting
After care will be the most important thing.
- Straight pumice, whether graded or not, is a common choice for soil
- Some people bare-root - I try to keep some old soil
- Regular misting of the foliage is very important
- Try not to keep the roots too wet
- Shade cloth and/or a greenhouse can help maintain humidity
It looks like the biggest challenge will be the logistics of actually getting it out without cutting too much. I'd want to follow lifelines to see which branches to keep and which to cut based on where the tree has roots.
Do let us know how it goes!