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 $\mathrm{\Omega }=\sum g_i{g}^i$ 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.