For a Lie algebra L with basis {x1,x2,…,xn}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.