It has been reported (Australian 26/6/19) that in August 2018 Malcolm Turnbull, at the time fighting for his political life, put it around that Peter Dutton, then his major opponent for the prime ministership, was ineligible to sit in the Parliament due to a possible conflict with section 44 of the Constitution and thus ineligible to be commissioned Prime Minister should he win the leadership ballot in the Liberal Party room. Mr Turnbull indicated that that would be his advice to the Governor-General.
Apparently, as also reported, the then Attorney General, Christian Porter, advised Mr Turnbull that the only consideration the Governor-General could give, should Mr Dutton win the ballot, was whether he could command a majority in the House of Representatives.
Mr Turnbull attempted to involve the Governor-General by inferring that he had spoken to him, a possible implication being he had his agreement. Obviously, he thought that by pushing this line, party members would decide not to vote for Peter Dutton. At that time, there was no thought that Scott Morrison would be a player in the leadership ballot.
Whilst the Governor-General, by convention, acts on the advice of the Prime Minister, he does not have to do so if he considers that that advice is bad and it was therefore wrong of Mr Turnbull to assume that the Governor-General would listen to him in this regard. The Governor-General is above party politics and any decisions made would be made by him alone, always in the interests of the people. He was, of course, free to call upon expert advice, as Sir John Kerr did in the political crisis of 1975. This also shows the measure of Malcolm Turnbull who apparently used the name of the Governor-General in an attempt to implicate him in his scheming to save his position.
Just imagine what the situation would have been had Australia become a republic with a president from a political party - “one of us” as republicans are wont to say. There is no telling whether a decision in such a matter would have been subjected to a bias rather than complete impartiality. Of course, Peter Dutton did not win the leadership ballot and the potential constitutional crisis Malcolm Turnbull was intent on causing was avoided.
Under our system of constitutional monarchy, the allegiance of the Governor-General is to the Queen and through the Queen to the people, never to politicians, not even to the prime minister nominated him or her. Under a republic there is no telling where allegiance will fall.