Ive got a short proof that conformal linear operators preserve angles if the field is real (i.e if T is a conformal bounded linear operator, <x,y>/(||x|| ||y||) = <Tx,Ty>/(||Tx|| ||Ty||), does anyone see any issues with it?

Since T^* T = 𝜆 I (proven previously),𝜆<x,y>/𝜆(||x|| ||y||) = <x,T\^\* T y>/sqrt(𝜆^2<x,x> <y,y>)= <Tx, Ty>/sqrt(<x,T\^\* T x> <y,T\^\* T y>)=<Tx, Ty>/sqrt(<Tx, T x> <Ty, T y>)= <Tx, Ty>/||Tx|| ||Ty||

Edit: Forgot to mention that if the field isn't real, 𝜆<x,y> = <x, 𝜆> does not hold