On the spectral set of a solvable Lie algebra of operators

Given $L$ a complex solvable finite dimensional Lie Algebra of operators acting on a Banach space $E$ and $\{x_i{\}}_{1\le i\le n}$ a Jordan-H"older basis of $L$, we study the relation between $Sp(L,E)$ and $\mathrm{\Pi }Sp(x_i)$.