Split 3-Leibniz algebras

We introduce and describe the class of split 3-Leibniz algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie triple systems and split 3-Lie algebras. More precisely, we show that any of such split 3-Leibniz algebras T is of the form T=U+∑jIj, with U a subspace of the 0-root space T0, and Ij an ideal of T satisfying [T,Ij,Ik]+[Ij,T,Ik]+[Ij,Ik,T]=0 for j≠k. Moreover, if T is of maximal length, we characterize the simplicity of T in terms of a connectivity property in its set of non-zero roots.