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Geodesic bases for Lie algebras

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and nonexistence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algebras having a codimension one ideal of a particular kind, nonunimodular algebras of dimension and unimodular algebras of dimension .