Jamieson, Fausset and Brown on 1 Kings 1:33

33. cause Solomon my son to ride upon mine own mule--Directions were forthwith given for the immediate coronation of Solomon. A procession was to be formed by the "servants of their lord"--that is, the king's bodyguard. Mules were then used by all the princes (2Sa 13:29); but there was a state mule of which all subjects were forbidden, under pain of death, to make use, without special permission; so that its being granted to Solomon was a public declaration in his favor as the future king (see on Es 6:8, 9).

bring him down to Gihon--a pool or fountain on the west of Jerusalem (see on 2Ch 32:30), chosen as equally public for the counter proclamation.