Commuting (centralizing) traces and Lie (triple) isomorphisms on triangular algebras revisited

The aim of the paper is to give a description of commuting (centralizing) traces of bilinear maps and Lie (triple) isomorphisms on certain triangular algebras. As consequences, commuting (centralizing) traces of bilinear maps and Lie (triple) isomorphisms on upper triangular matrix algebras over a 2-torsion free commutative ring are completely determined.