It doesn't matter what basis is used to define Casimir, as long as the Killing form is used properly: In any simple Lie algebra, by Cartan's criterion the Killing form $\langle,\rangle$ is non-degenerate. Given any basis $x_i$ for the Lie algebra, let x_i' be the dual basis with respect to $\langle,\rangle$. Then the Casimir element is \sum_i x_i x_i' in the universal enveloping algebra.
Many sources give an exercise to prove that this expression is independent of basis and that it is central in the enveloping algebra, but there is a direct argument (probably found in Serre, for example): the natural maps with simple Lie algebra $g$, $End(g) \approx g \otimes g^* \approx g\otimes g \subset \bigotimes{}^\bullet g \rightarrow Ug$ are G-equivariant, where the identification of $g^*$ with $g$ is exactly via the Killing form. The identity map among endomorphisms of $g$ commutes with $G$, so its image at the other end does, as well. Consideration of the second and third items yields expressions as in the first paragraph. Poincare-Birkhoff-Witt proves that the resulting expressions are non-zero.