  Let $\{C_i\}_{i\in \Gamma_1},$ and $\{D_j\}_{j\in \Gamma_2},$ be two families
of Cartan factors such that all of them have dimension at least $2$, and
consider the atomic JBW$^*$-triples $A=\bigoplus\limits_{i\in
\Gamma_1}^{\ell_{\infty}} C_i$ and $B=\bigoplus\limits_{j\in
\Gamma_2}^{\ell_{\infty}} D_j$. Let $\Delta :A \to B$ be a
{\rm(}non-necessarily linear nor continuous{\rm)} bijection preserving the
truncation of triple products in both directions, that is, $$\begin{aligned}
  \boxed{a \mbox{ is a truncation of } \{b,c,b\}} \Leftrightarrow
\boxed{\Delta(a) \mbox{ is a truncation of } \{\Delta(b),\Delta(c),\Delta(b)\}}
  \end{aligned}$$ Assume additionally that the restriction of $\Delta$ to each
rank-one Cartan factor in $A$, if any, is a continuous mapping. Then we show
that $\Delta$ is an isometric real linear triple isomorphism. We also study
some general properties of bijections preserving the truncation of triple
products in both directions between general JB$^*$-triples.
