  Let $\mathcal{T}$ be a triangular algebra over a commutative ring
$\mathcal{R}$ and $\mathcal{Z(T)}$ be the center of $\mathcal{T}$. Suppose that
${\mathfrak q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}$
is an $\mathcal{R}$-bilinear mapping and that ${\mathfrak T}_{\mathfrak
q}\colon: \mathcal{T}\longrightarrow \mathcal{T}$ is a trace of $\mathfrak{q}$.
We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition
$[{\mathfrak T}_{\mathfrak q}(T), T]\in \mathcal{Z(T)}$ for all $T\in
\mathcal{T}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper
form will be addressed. Using the aforementioned trace function, we establish
sufficient conditions for each Lie triple isomorphism on $\mathcal{T}$ to be
almost standard. As applications we characterize Lie triple isomorphisms of
triangular matrix algebras and nest algebras. Some further research topics
related to current work are proposed at the end of this article.
