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 = U{script} + Σj Ij with U{script} a subspace of the 0-root space T0 and any Ij a well described ideal of T, satisfying [Ij, T, Ik] = 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. © Printed in India.