Spectral invariants for finite dimensional Lie algebras

For a Lie algebra $\mathcal{L}$ with basis $\{x_1,x_2,\cdots ,x_n\}$, its associated characteristic polynomial $Q_{\mathcal{L}}(z)$ is the determinant of the linear pencil $z_{0}I+z_1\text{ad}{x}_1+\cdots +z_n\text{ad}{x}_n\mathrm{.}$ This paper shows that $Q_{\mathcal{L}}$ is invariant under the automorphism group $\text{Aut}(\mathcal{L})\mathrm{.}$ The zero variety and factorization of $Q_{\mathcal L}$reflectthestructureof${\mathcal L}$\mathrm{.}Inthecase${\mathcal L}$ is solvable $Q_{\mathcal{L}}$ is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincar'{e} polynomial for solvable Lie algebras. Application is given to $1$-dimensional extensions of nilpotent Lie algebras.