Normal operator equal to its norm and that of its adjoint

Question

I am coming up with this equality over and over again in Linear ALgebra Done Right by Axler and I cant seem to derive it. I've tried looking up something similar here, the closest I've got is this:Normal operator iff norm on v equivalent to that of adjoint , but the equality is only stated. I understand both $TT^*$ and $T^*T$ are self-adjoint, as well as T normal but I don't see how this implies $\langle TT^*v,v\rangle = \langle T^*v,T^*v\rangle$, and $\langle T^*Tv,v\rangle = \langle Tv,Tv\rangle$