On the structure of graded Leibniz triple systems

We study the structure of a Leibniz triple system $\mathcal{E}$ graded by an arbitrary abelian group $G$ which is considered of arbitrary dimension and over an arbitrary base field $\mathbb{K}$. We show that $\mathcal{E}$ is of the form $\mathcal{E}=U+{\sum }_{[j]\in {\sum }^1\mathrm{/}\sim }{I}_{[j]}$ with $U$ a linear subspace of the 1-homogeneous component ${\mathcal{E}}_1$ and any ideal $I_{[j]}$ of $\mathcal{E}$, satisfying ${I_{[j]},$\mathcal{E}$,I_{[k]}} ={I_{[j]},I_{[k]},$\mathcal{E}$}={$\mathcal{E}$,I_{[j]},I_{[k]}}=0$if$[j]\neq [k]$,wheretherelation$\sim$in$\sum^{1}={g \in G \setminus {1} : L_{g}\neq 0}$,definedby$g \sim h$ifandonlyif$g$isconnectedto$h$.