On split Lie triple systems

We begin the study of arbitrary split Lie triple systems by focussing on those with a coherent 0-root space. We show that any such triple systems T with a symmetric root system is of the form T = + Μ _j I _j with a subspace of the 0-root space T ₀ and any I _j a well described ideal of T, satisfying [I _j , T, I _k ] = 0 if j ≠ k. Under certain conditions, it is shown that T is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. The key tool in this job is the notion of connection of roots in the framework of split Lie triple systems.