An Adjoint Criterion for Left-Invariant $m$-Modified Conformal Vector Fields on Even-Dimensional Lie Groups

For a fixed positive real constant $m$, a real Lie algebra of even dimension $2k$ admits a nonzero left-invariant $m$-modified conformal vector field for some nondegenerate metric if and only if an adjoint operator has a nonzero real eigenvalue. In that case, a solution exists with neutral signature