In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.

In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank at least 4 with an idempotent are Jordan algebras.

Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of e-stable power-associative algebras.

We also prove that the algebra obtained by 2 -gametization process of a Lie triple algebra is a Lie triple one.

Lie triple algebra; Pseudo-idempotent; Jordan algebra; Peirce decomposition; Train algebra