It's not about law in the sense of criminal law.
You used the analogy of a pirate crew. Lets assume some officers of that pirate ship planned a mutiny. The second in command kills the offenders without the captain demanding their heads.
The captain could claim that the second in command acted on his/her own will. But this would imply that the second in command acted without considering the will of the captain to be important. The captain looks weak as he/she got ignored by the second in command and didn't have the total control over the situation.
Doing something to be able to deny it later on is weak as it gives the impression that the captain him-/herself isn't confident in the decision. If the fate of everybody is depending on the swift and good judgment of situations from the captain in charge anything that looks like making decisions that even the captain isn't sure about is a really bad sign.
Any ruler, who rules by force/will, absolutely needs to make perfectly clear that he/she is in complete control of the situation. Failing to do so opens up possibilities, possibilities that ruler surely doesn't want.
Display of power needs to have a very clear message. Not only is it important that the actions have the right results, they need to send the right messages too.
The Emperor now looks weak. Sure, the traitors got the punishment they deserved. But it wasn't the clear will of the Emperor that resulted in their deaths. It was Patreus who did what had to be done.
And speaking of clear messages, killing somebody without the traditional show trial sends the very clear message that those who got killed where to dangerous to stay alive. And again the Emperor looks weak.
At the end of the day we have:
Patreus being the strong man with a clear vision and doing what had to be done.
The traitors being dead, but the Empire acknowledging that they posed an essential threat to the Emperor.
The Emperor did nothing to solve the crisis.
And we all know that killing a criminal only sends one message to other criminals. It's not "don't be a criminal", it's "Don't get caught".
(I'm curious how FDev is going to continue that story line - if at all.)