# Derivations and dimensionally nilpotent derivations in Lie triple algebras

## TL;DR

This paper investigates derivations in non-nilpotent Lie triple algebras, characterizing their structure based on idempotent elements and exploring their multiplicative properties, including connections to gametic and Bernstein algebras.

## Contribution

It provides a detailed classification of derivation structures in non-nilpotent Lie triple algebras and links their basis structures to known algebraic forms.

## Key findings

- Derivation algebra structure depends on the presence of idempotents.
- When dimension n=2p+1, bases align with gametic algebra G(2p+2,2).
- For n=2p, the algebra is either like the previous case or a Bernstein algebra.

## Abstract

In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether the algebra admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nilpotent dimensionally nilpotent Lie triple algebras. We show that when $n=2p+1$ the adapted basis coincides with the canonical basis of the gametic algebra $G(2p+2,2)$ or this one obviously associated to a pseudo-idempotent and if $n=2p$ then the algebra is either one of the precedent case or a conservative Bernstein algebra.