‎Beside the triple product induced by ultrapowers on the bidual of a JB$^*$-triple‎, ‎we assign a triple product to the bidual‎, ‎$E^{**},$ of a JB-triple system $E,$ and we show that‎, ‎under some mild conditions‎, ‎it makes $E^{**}$ a JB-triple system‎. ‎To study ternary $n$-weak amenability of $E^{**},$ we need to improve the module structures in the category of JB-triple systems and their iterated duals which lead us to introduce a new type of ternary module‎. ‎We then focus on the main question‎: ‎when does ternary $n$-weak amenability of $E^{**}$ imply the same property for $E?$ In this respect‎, ‎we show that‎, ‎if the bidual of a JB$^*$-triple $E$ is ternary $n$-weakly amenable then $E$ is ternary $n$-quasi-weakly amenable‎. ‎For a general JB-triple system‎, ‎however‎, ‎the results are slightly different for $n=1$ and $n\geq2$‎, ‎and the case $n=1$ requires some additional assumptions‎.

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