• rage comes from Old French rage, from Late Latin rabia, a regularization of Classical Latin rabies.
• lobby and lodge are both from Late Latin laubia, a loanword from Frankish laubija, meaning “arbour” or “shelter”.
• sage (the plant) comes directly from its Latin name salvia. (As a sidenote, the psychedelic salvia is one of many salvia/sage species and is native to Mexico).
• rouge is from Latin rubeus (“red”), while ruby is from Latin rubinus. This means that while ruby and rouge are closely related, they’re not a neat doublet like the others.
• The latter half of (sub)servient and sergeant are both from Latin servientem meaning “serving”.

Other pairs include sapient/sage (meaning wisdom, as in sage advice) and cavity/cage (the latter from Latin cavea > *cavia).