. . I never understood why Forrest didn't throw Archer out the airlock after he regained command. Or for that matter, why Archer didnt kill Forrest in the first place. . .
The only thing stopping Forrest was Starfleet by then had received Archer's previously sent "insurance policy", and had ordered Forrest to investigate. He does give T'Pol orders for disposal before she boards Defiant.
It seems the only reason for the other matter was the writers wanted Forrest alive so he could be killed when Enterprise blows up in Part One.