On split Lie triple systems (Q841195)

A Lie triple system \(T\) is said to be split if there is a maximal abelian diagonalizable subalgebra \(H\) of its Lie algebra of inner derivations. In this situation, the triple system splits as \(T=T_0\oplus(\oplus_{\alpha\in\Lambda}T_\alpha)\), for a set \(\Lambda\) of nonzero linear functions on \(H\), where \(T_\alpha=\{ x\in T: d(x)=\alpha(d)x\;\forall d\in H\}\). The paper under review is devoted to study these systems under some extra technical conditions on the root system \(\Lambda\), which guarantee that any such Lie triple system is a direct sum of simple ones. The key tool is a notion of connection of roots in this setting.