Commutativity-preserving operators on symmetric matrices (Q798732)

In a recent paper \textit{G. H. Chan} and \textit{M. H. Lim} [ibid. 47, 11-22 (1982; Zbl 0492.15006)] show that any nonsingular linear operator on the real space of symmetric matrices which preserves commutativity must be of a specific form. In the present paper the author shows that essentially the same result holds if the ground field is taken to be formally real; that is, fields in which 0 is not the sum of squares of nonzero elements. Some closing remarks discuss the modification necessary in the form of the linear transformation in the formally real case and the possibility of extending the result to other fields.