Commutativity-preserving operators on symmetric matrices

If φ is a nonsingular linear operator on n × n symmetric matrices over a formally real field, n ? 3, and if φ preserves commutativity, then φ(A) = cU^-1AU + f(A)I, where U^tU and c are scalar and ? is a linear functional. This is an extension of a result of Chan and Lim.