Isometries of real Hilbert C⁎-modules

Let T:V→WT:V→W be a surjective real linear isometry between full real Hilbert C⁎C⁎-modules over real C⁎C⁎-algebras A and B, respectively. We show that the following conditions are equivalent: (a) T is a 2-isometry; (b) T is a complete isometry; (c) T preserves ternary products; (d) T preserves inner products; (e) T is a module map. When A and B are commutative, we give a full description of the structure of T.