I'm sorry to bother but i'm having some problems in proving that, given a simple Lie Algebra L of finite dimension $n$ (equipped with the Killing form) and its enveloping universal algebra U(L), then the element (Casimir): c = $\sum x_iy_i$ where $(x_i)_i$ is a basis and $(y_i)_i$ is its dual basis (with respect to the Killing form) doesn't depend on the choice of a particular basis.
The argument should be just related to linear algebra i guess. I tried to take another basis and the change-of-coordinates-matrix but i can't get to the solution.
Can someone help me? Am I missing something? Thank you