Lorentz transformation, quantum field theory

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Hello, I was reading and trying to follow up with Pierre Ramond's "Field theory: A modern primer" and got stuck in his step to step jumping. Kindly, see attachment and note that Eq (1.2.6) = g_{ρσ}=g_{μ\upsilon}\Lambda^{μ}_{ρ}\Lambda^{\upsilon}_{σ}.

My question is what do I need from tensor calculus to get how did he jump between (1.2.20), (1.2.21), (1.2.22)? Thank you
 

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Just knowledge on how to manipulate contracted products.
 
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