'Reign,' 'royal,' 'regent,' and 'regulate' all share PIE *h3reg- (to direct, to rule). One root, all power.
(noun) The period during which a sovereign rules; royal authority. (verb) To rule as sovereign; to be predominant or prevalent.
From Old French 'reigne' (kingdom, rule), from Latin 'rēgnum' (royal power, kingdom, dominion), from 'rēx' (king, genitive 'rēgis'), from 'regere' (to rule, to guide, to make straight). The PIE root is *h₃reǵ- (to move in a straight line, to direct, to rule), the same root behind 'regal,' 'royal,' 'rector,' 'regime,' and — via Sanskrit 'rājan' (king) — the Indian royal title 'raja.' The silent 'g' in English 'reign' reflects the Latin 'g' in 'rēgnum,' preserved in spelling long after French phonology
English has three homophones — 'reign,' 'rain,' and 'rein' — all pronounced /ɹeɪn/ but from completely different origins. 'Reign' is from Latin 'rēgnum' (kingdom), 'rain' from Old English 'regn' (a Germanic word), and 'rein' from Old French 'rene' (a strap). The phrase 'free rein' is often misspelled as 'free reign,' conflating the horse metaphor with royal power.