(Drive 2011) Why did the Driver stomp out the assassin in the elevator so brutally?

In Drive, one of the most famous and pivotal scenes in the film shows the protagonist beat up, and brutally pulverise an assassin to save the person he loves. Before he does, he kisses her, representing the idea that after she sees him kill that man, she could never see him in the same way again. My question is, why stomp his head into a pulp? Why wouldn't he act in "self defense" and handle the situation a little better so that he could keep his love interest? Clearly the driver knew this, but proceeded to cave his head in anyway.

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