Summary: A naturally parameterised curve in a Lie group with a left invariant metric is a geodesic, if its tangent vector left-translated to the identity satisfies the Euler equation \(\dot{Y} = \operatorname{ad}^t_Y Y\) on the Lie algebra \(\mathfrak{g}\) of \(G\). Stationary points (equilibria) of the Euler equation are called geodesic vectors: the geodesic starting at the identity in the direction of a geodesic vector is a one-parameter subgroup of \(G\). We give a complete classification of Lyapunov stable and unstable geodesic vectors for metric Lie algebras of dimension 3 and for unimodular metric Lie algebras of dimension 4.