On the existence of orthonormal geodesic bases for Lie algebras

TL;DR

This paper investigates the existence of orthonormal geodesic bases in unimodular Lie algebras, proving their existence in dimensions up to 4 and providing a counterexample in dimension 5.

Contribution

It establishes the existence of orthonormal geodesic bases for unimodular Lie algebras up to dimension 4 and shows such bases may not exist in dimension 5.

Findings

Orthonormal geodesic bases exist for unimodular Lie algebras of dimension ≤ 4.

Counterexample of a 5-dimensional unimodular Lie algebra without such a basis.

Existence depends on the algebra's dimension and structure.

Abstract

We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product.