  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 non-existence 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 \$\leq 4\$, and unimodular algebras of
dimension 5.
