Writing something in the king's name represents writing it with his authority, or writing it as his representative. (See: [[rc://en/ta/man/translate/figs-metonymy]])
The royal stud was the king's best male horse. The horses that were used in the king's service were its offspring. AT: "the offspring of the king's best horse" (See: [[rc://en/ta/man/translate/figs-explicit]])