Equivalent characterizations of a finite-dimensional isometry

Let \(V\) be a finite-dimensional real inner product space with inner product \((\ ,\ )\), and let \(T:V\to V\) be linear. Show that the following are equivalent:

1. \((Tx,Ty)=(x,y)\) for all \(x,y\in V\);

2. \(\|T(x)\|=\|x\|\) for all \(x\in V\);

3. \(T^*T=\operatorname{Id}_V\), where \(T^*\) is the adjoint of \(T\);

4. \(TT^*=\operatorname{Id}_V\).