is he really trying to kill the 2nd prince or the opposite? It seems to me he is trying to make the 2nd prince emperor while cleaning up the noble faction.
The 2nd prince does not want to be emperor, so if he makes him crown prince he would decline. But by making the 3rd prince emperor, the 2nd prince has to kill the 1st prince.
meanwhile 2nd prince is building foundation with commoners and lower nobles, while the emperor has the higher nobles murder each other in this war.