By your last phrase where does it stay the affirmation "he's lawful withing his organization (even if we admit his organization is outlawed by general rules)"...this was my point.
I admit he may be a little bit crazy

but not at all a case of split personality cause he's fully aware of those two aspects all the time always judging if something has anything to do with his organization's goals and absolutely never his good side takes over when on duty (he doesn't go on a rampage unnecessarily) neither the evil side in his free time.
I guess I see where you are going with this. "Lawful," does not necessarily mean that he always abides by the ultimate rules of his society just that he recognizes some kind of order and subscribes to it. He may disagree with the established order on moral grounds (Stormcloaks) or he may be motivated by less than altruistic desires (Dark Brotherhood), but he does not go out of his way to create discord.
For instance, using the DB as an example.
LE - would complete jobs respect his "brothers," and do what he could to advance the organization. If there was a power struggle he would try to gain consensus among the members and advance democratically. He sees his organization's role in the world as necessary which is why he does not oppose it's existence in a society where it is outlawed. If there was a power struggle he would support whoever he had pledged his fealty to though he might not work too hard to save them if he found himself on the losing side.
NE - is purely motivated by his own ends. Power, money, fame, etc. He will do whatever it takes to achieve them. He has no friends in the organization. They just exist as pawns for him to move around on his chess board. If there was a power struggle he would side with whichever faction served his purposes until they no longer served his purposes. He would have no qualms about killing his superiors, servants, etc. as long as it helped him achieve his goals.
CE - is unpredictable and brutal. He enjoys killing for it's own sake and is more of a serial killer the fact that he gets paid to do it is incidental. He would thrive in an environment where there was a power struggle and might just keep on killing until everyone was dead. He would feel no guilt assassinating his superior and would only hold off if his superior was too powerful to get at. He started making plans to kill everyone the moment he joined the organization.