On symmetry of the (strong) Birkhoff-James orthogonality in Hilbert C*-modules

In this note, we prove that the Birkhoff-James orthogonality, as well as the strong Birkhoff-James orthogonality, is a symmetric relation in a full Hilbert A-module V if and only if at least one of the underlying C*-algebras A or K(V) is isomorphic to C.