Idempotents play an important role in the investigation of nonassociative algebras structure ([2],[18],,[21]). However, the existence of such elements is not always guaranteed, specially when dealing with algebras that are defined by polynomial identities. Hence, in many cases, one has to assume the existence of idempotents $([3,11,17])$ in order to investigate algebraic structures via the classical Peirce decomposition.
Lie triple algebras appear historically and independently in [11] and [16].
In this paper, we deal mainly with Lie triple algebras which admit no idempotent. We prove that non nil algebras possess either an idempotent or a pseudo-idempotent. In the latter case, it is possible to obtain a Peirce decomposition relatively to the pseudo-idempotent and some relations on Peirce's components. We notice that these results are analogous to that obtained by Osborn in ([11]). When a Lie triple algebra has a non-trivial idempotent, we show that its heart is a Jordan subalgebra. Then we give a Lie triple non-conservative algebras classification for dimensions 2 and 3. Finally, we characterize train algebras of rank at least that are Lie triple algebras.

Lie triple algebra pseudo-idempotent Jordan algebra Peirce decomposition