Conformal Killing Symmetric Tensors on Lie Groups

We introduce the notion of metric Lie algebras of Killing type, which are charac- terized by the fact that all left-invariant 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.