'Seat' is Old Norse from PIE *sed- (to sit) — the root of 'sit,' 'settle,' 'session,' and 'president.'
A thing made or used for sitting on; a place in which someone sits; to cause to sit down; to install in a position of authority.
From Old Norse sæti (a seat, position, place of residence), from Proto-Germanic *sētiz (a seat, a settled place), from PIE *sed- (to sit). The PIE root *sed- is one of the most productive in Indo-European, generating sit, set, settle, saddle, and session in English alone, plus Latin sedēre (to sit), which gives sediment (material that has settled), sedentary (sitting-habitual), preside (to sit before, to lead), reside (to sit back, to dwell), subside (to sit down), obsess (to sit upon), and sedate (to calm into sitting). Old English had the cognate
A 'seat of government' and a 'seat in parliament' both preserve the ancient connection between sitting and authority. In Latin, the same root produced 'sēdēs' (a seat, a throne, an episcopal see), which is why a bishop's official church is a 'cathedral' — from 'cathedra' (a chair, from Greek kata- 'down' + hedra 'seat,' from the same PIE *sed-). Authority in the Indo-European world was