On the structure of graded Leibniz triple systems

We study the structure of a Leibniz triple system EE graded by an arbitrary abelian group G   which is considered of arbitrary dimension and over an arbitrary base field KK. We show that EE is of the form E=U+∑[j]∈∑1/∼I[j]E=U+∑[j]∈∑1/∼I[j] with U   a linear subspace of the 1-homogeneous component E1E1 and any ideal I[j]I[j] of EE, satisfying {I[j],E,I[k]}={I[j],I[k],E}={E,I[j],I[k]}=0{I[j],E,I[k]}={I[j],I[k],E}={E,I[j],I[k]}=0 if [j]≠[k][j]≠[k], where the relation ∼ in ∑1={g∈G∖{1}:Lg≠0}∑1={g∈G∖{1}:Lg≠0}, defined by g∼hg∼h if and only if g is connected to h.