This is a regular proper surface over R but is not in general smooth over R or a group scheme over R. Its subscheme of smooth points over R is the Néron model, which is a smooth group scheme over R but not necessarily proper over R. The fibers in general may have several irreducible components, and to form the Néron model one discards all multiple components, all points where two components intersect, and all singular points of the components.
This surface is an ordinary regular surface over R but lacks general smoothness over R or as a group scheme over R. Its subset of smooth points forms the Néron model, which is a smooth group scheme over R but may not be proper over R. The fibers typically consist of multiple irreducible components, and to form the Néron model, one must eliminate all multiple components, intersection points of two components, and singular points.