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 ≤ 4, and unimodular algebras of dimension 5.

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra g (respectively, of the Grassmannian of two-planes of g) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on g. In the second part we study the subsets of g which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closures of these subsets is the whole algebra g, apart from two exceptional cases: when g is two-step nilpotent and when g contains a codimension one abelian ideal.

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