Very nilpotent basis and n-tuples in Borel subalgebras

TL;DR

This paper characterizes nilpotent Lie algebras through the concept of very nilpotent bases and explores the structure of n-tuples in Borel subalgebras within semisimple Lie algebras.

Contribution

It refines Engel's Theorem by establishing that a Lie algebra has a very nilpotent basis if and only if it is nilpotent, and describes the algebraic set of n-tuples in Borel subalgebras.

Findings

A Lie algebra has a very nilpotent basis iff it is nilpotent.

Defined an ideal related to n-tuples in Borel subalgebras.

Characterized the set of n-tuples in Borel subalgebras for semisimple Lie algebras.

Abstract

A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S((g^n)^*)^G whose associated algebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.