Adam Clarke on 1 Kings 1:33

Take with you the servants of your lord - By these we may understand the kings guards, the guards of the city, the Cherethites and Pelethites, who were under the command of Benaiah; and in short, all the disposable force that was at hand.

Solomon - to ride upon mine own mule - No subject could use any thing that belonged to the prince, without forfeiting his life. As David offered Solomon to ride on his own mule, this was full evidence that he had appointed him his successor.