Rank-one nonincreasing additive mappings on second symmetric product spaces

In this note we describe those additive mappings from a second symmetric product space to another, over a field of characteristic not 2 or 3, which preserve decomposable elements of the form λu · u where u is a vector and λ is a scalar. This leads to the corresponding result concerning additive mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one. We also discuss some consequences of this characterization theorem.