  We introduce the notion of metric Lie algebras of Killing type, which are
characterized by the fact that all conformal Killing symmetric tensors are sums
of Killing tensors and multiples of the metric tensor. We show that if a Lie
algebra is either 2-step nilpotent, or 2- or 3-dimensional, or 4-dimensional
non-solvable, or 4-dimensional solvable with 1-dimensional derived ideal, or
has an abelian factor, then it is of Killing type with respect to any positive
definite metric.
