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