superintend, manager, rule
宰 centers on authoritative control: superintending, managing, or ruling over affairs or people.
宰 combines 宀 (roof/building) with 辛, which likely contributes the sound and may relate to a sharp tool or authority. The character suggests someone who exercises control within a domain.
Under a roof 宀, a sharp tool 辛 represents the authority of a manager who superintends and rules.
For サイ, imagine a manager sighing as they superintend a busy office: sigh -> サイ.
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chairmanship; presidency; management
prime minister; premier; chancellor
dazaifu (under the ritsuryō system, governmental office with jurisdiction over Kyushu, Iki and Tsushima)
president; chairman
incompetent cabinet minister; figurehead minister
Minister of State (Zhou dynasty China)
chef; head cook
government official working for the daizaifu
director of the dazaifu (ritsuryō system)
supervision; superintendence; management; supervisor
(electoral) campaign manager
Buddhist priest who is also an important politician