Lorentz Transformation: Conservation of Inner Product?

[princeton118]

Is Lorentz transformation one kind of coordinate transformation? If so, the conservation of inner product, which is the primary property of Lorentz Transformation, is trivial, isn't it? The vector transforms in some way, the metric transforms in the inverse way, so the inner product defintely will not change under a coordinate transformation. If so, any coordinate transformation can be Lorentz transformation?

 

[robphy]

The transformation must be linear as well.