Writing something in the king's name represents writing it with his authority, or writing it as his representative. (See: [[rc://en/ta/man/jit/figs-metonymy]])
You may need to use a word for written instructions that is more general than "letter" so that the reader does not think that these documents were about only personal matters.
The royal stud was the king's best male horse. The horses that were used in the king's service were its offspring. Alternate translation: "the offspring of the king's best horse" (See: [[rc://en/ta/man/jit/figs-explicit]])