Albert Barnes Commentary 1 Kings 1:33

"And the king said unto them, Take with you the servants of your lord, and cause Solomon my son to ride upon mine own mule, and bring him down to Gihon:" — 1 Kings 1:33 (ASV)

Mules and horses seem to have been first employed by the Israelites in the reign of David, and their use was at first confined to prominent people (2 Samuel 13:29; 2 Samuel 18:9). The rabbis tell us that it was a capital offense to ride on the king’s mule without his permission; this would have made it all the more evident to everyone that the events concerning Solomon had David’s sanction.

Gihon – This was probably the ancient name for the valley later called the Tyropoeon, which ran from the present-day Damascus Gate, past Siloam, and into the Kidron Valley. This valley had the Temple Mount, or the true Zion, on its left, and the modern Mount Zion, or the ancient city of the Jebusites, on its right. The upper “source” of the “waters of Gihon,” which Hezekiah stopped, was probably near the Damascus Gate.