Although nobody says it like that, Tywin has been the real head of the House Lannister since the beginning of the show.
He rarely confronts people directly, he almost never does the dirty work, but by the end of the day, Tywin is the guy who gets things done.
The Lannister grandpa has his hands deep in everyone’s business, but his most important connection is the one he has with his son Tyrion, whom he detests and respects at the same time.
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This is probably why everyone was shocked when the Imp killed his dad. With a crossbow. While the man was on the toilet.