system
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human
[Star Wars] Was Vader ever going to try and kill palpatine?

Unlike most Sith who want to be more powerful and in charge or whatever Vader seems to lack ambition. At least from what I've seen his plan to overthrow the Emperor and rule the galaxy died with Padme(until Luke). Now he just wants to hurt people like he is also hurting. 

A bonus question, what would Palpatine do if Vader never made a move on him? Would he eventually be frustrated by his apprentice's lack of drive? Obviously no Sith actually wants to be killed by his apprentice but having someone nipping at your toes forces you to evolve and improve as well. 

An apprentice who is too strong to be killed by any potential rivals for you to replace them with but who also lacks any drive to come at you seems kind of disappointing and liable to lead to stagnation.

I know he was researching force cloning and stuff but I don't think that necessarily means he had abandoned the rule of two. All Sith seek to further dark knowledge in such ways as to become THE Sith Lord and don't believe they will ever be killed by their apprentice. The guy that invented force lightning probably thought he could never be defeated but he still took another apprentice who eventually defeated him and took that knowledge further.
gpt
Vader did basically lose most drive to kill Palpatine. After losing Padme and his limbs, Sidious was all he had left. His only remaining friend who found value in him. He was content because there was nothing else. Turning to the dark side was for Padme and without her his ambition did take a hit. Not that he never thought about it. But he was overall content. His desires were mostly fleeting moments of frustration and plans that fell through. 

That is until he learns he has a son. That returns his drive because now he has something. He has a family again and with his son having his potential he could overthrow the Emperor far more easily. Luke basically changes his life by existing. 

Well sending assassins and threatening to replace him was basically how Vader was kept oh his toes, along with him continually pushing him through trials. Palpatine is well aware of their relationship and is fine with Vader as he is because his potential is always dangerous but he definitely doesn't foresee himself losing. And he would replace Vader with a more worthy candidate at a drop like he tried with Luke.