Casimir invariant for an invariant nondegenerate bilinear form on a finite-dimensional Lie algebra

Definition
Suppose $$L$$ is a finite-dimensional (say $$n$$-dimensional) Lie algebra over a field $$F$$ and suppose $$b$$ is an invariant (under automorphisms of $$L$$) nondegenerate bilinear form on $$L$$. The Casimir invariant or Casimir element for $$L$$ with respect to $$b$$ is an element in the center of the universal enveloping algebra $$U(L)$$ defined as follows:


 * First, pick a basis $$x_1,x_2,\dots,x_n$$ for $$L$$ as a vector space over $$F$$.
 * Now, pick a dual basis for $$L$$ with respect to the original basis for the bilinear form $$b$$. Note that this is possible because $$b$$ is nondegenerate. Call the dual basis $$y_1,y_2,\dots,y_n$$. What this means is that $$b(x_i,y_i) = 1$$ for all $$i$$ and $$b(x_i,y_j) = 0$$ for $$i \ne j$$.
 * Define the Casimir invariant for $$b$$ as the following element $$\Omega$$ of $$U(L)$$:

$$\Omega = \sum_{i=1}^n x_iy_i$$

Interpretation as an operator
If the Lie algebra $$L$$ acts as derivations on some structure, then the corresponding universal enveloping algebra acts as differential operators on the same structure. In this context, Casimir elements act as differential operators. Hence, we often use the term Casimir operator.

Facts in the definition

 * Casimir invariant is independent of choice of basis: The value of the Casimir invariant depends only on the Lie algebra and the bilinear form, not on the choice of basis.
 * Casimir invariant is central: The Casimir invariant is in the center of the universal enveloping algebra.

Particular cases
The typical setup for taking the Casimir invariant is a semisimple Lie algebra and the Killing form on it. Note that, by Cartan's second criterion, if $$F$$ is algebraically closed, the Killing form must be nondegenerate. There are many cases where the Killing form is nondegenerate even for $$F$$ not algebraically closed.