particle N1 uncommon written
〜下で — under
〜下で ・ かで
A full written guide for 〜下で is on the way.
This point is mapped in the grammar tree, but its examples, formation table and usage notes haven't been written yet. The fastest way to learn it now is to meet it in real sentences inside Jengo.
See 〜下で in real sentences
Jengo shows 〜下で the way you actually meet it: inside real Japanese sentences, so it sticks instead of staying an abstract rule.
Study it in JengoSources Compiled with reference to A Dictionary of Intermediate Japanese Grammar. Editorial confidence: low.