Casimir operator

Let $\g$ be a semisimple Lie algebra, and let $(\cdot,\cdot)$ denote the Killing form. If $\{g_i\}$ is a basis of $\g$ , then there is a dual basis $\{g^i\}$ with respect to the Killing form, i.e., $(g_i,g^j)=\delta_{ij}$ . Consider the element $\Omega=\sum g_ig^i$ of the universal enveloping algebra of $\g$ . This element, called the Casimir operator is central in the enveloping algebra, and thus commutes with the $\g$ action on any representation.