Centralizing traces with automorphisms on triangular algebras

Let T be a triangular algebra over a commutative ring R, ξ be an automorphism of T and Z_ξ(T) be the ξ-center of T. Suppose that q: T× T⟶ T is an R-bilinear mapping and that T_q: T⟶ T is a trace of q. The aim of this article is to describe the form of T_q satisfying the commuting condition [Tq(x),x]ξ=0 (resp. the centralizing condition [Tq(x),x]ξ∈Zξ(T)) for all x∈ T. More precisely, we will consider the question of when T_q satisfying the previous condition has the so-called proper form.