1. under (guidance, supervision, rules, the law, etc.)
oft. as ...の下に or ...の下で
あいつは今、誰の下で働いてる？Who's that guy working under now?
2. under (a flag, the sun, etc.); beneath
彼は太陽の下で笑っていた。He was smiling under the sun.
3. with (e.g. one blow); on (the promise, condition, assumption, etc. that ...); in (e.g. the name of ...)
as ...の下に
4. (someone's) side; (someone's) location
usu. written as 元