Quote 1
ANTONIO
We all were sea-swallowed, though some cast again,
And by that destiny to perform an act
Whereof what's past is prologue, what to come
In yours and my discharge. (2.1.287-290)
Being sea-swallowed and removed from the court gives the traitors the freedom to do what would otherwise be an unthinkable act. The island has no rules.
Quote 2
ANTONIO
Let's all sink wi' th' king.
SEBASTIAN
Let's take leave of him. (1.1.66-67)
From the first sign that things might not be rosy, Sebastian shows he has no loyalty to his brother or his king (Alonso is both). He has a partner in callousness in Antonio—this foreshadows their later treachery on the island.
Quote 3
ANTONIO
Nor I. My spirits are nimble.
They fell together all, as by consent.
They dropped as by a thunderstroke. What might,
Worthy Sebastian? O, what might—? No more.
And yet me thinks I see it in thy face
What thou shouldst be. Th' occasion speaks thee, and
My strong imagination sees a crown
Dropping upon thy head.
SEBASTIAN
What, art thou waking?
ANTONIO
Do you not hear me speak?
SEBASTIAN
I do, and surely
It is a sleepy language and thou speak'st
Out of thy sleep. (2.1.224-236)
The betrayal Antonio suggests is so heinous as to be unfathomable to Sebastian at first. It seems that once you gain something by betrayal, you're willing to do it over and over again, because it works so well… until it doesn't. (Think of Macbeth's gains and downfall here.)