What part of the brain would it be fatal to injure?

Brain injury: Brain stem injury could be fatal. Brain stem includes various vital center which maintain hear rate, respiration, arousal system , center for eye movements, which all can be affected by brainstem injury

Can be variable: Obviously, brainstem injury could be instantaneously fatal, but could be primary problem in higher brain areas such as bleeding, subdural hematoma causing secondary brain herniation which could result in permanent damage.