Geodesic vectors and flat totally geodesic subalgebras in nilpotent metric Lie algebras

We determine geodesics and flat totally geodesic subalgebras in higher-step nilpotent metric Lie algebras of dimension $5$. It is surprising that in nonfiliform metric Lie algebras with one-dimensional center, the geodesic vectors and flat totally geodesic subalgebras are independent of the choice of the inner product.