Proposition 11

*Let $g$ be a Lie algebra over a field $K$, $U$ its enveloping algebra, $h$ a finite-dimensional ideal of $g$ and $\beta$ an invariant bilinear form on $g$, whose restriction to $h$ is non-degenerate. Let $(e_i)_{1 \leq i \leq n}, (e'_j)_{1 \leq j \leq n}$ be two bases of $h$ such that $\beta(e_i, e'_j) = \delta_{ij}$. Then the element $c = \sum_{i=1}^n e_i e'_i$ of $U$ belongs to the centre of $U$ and is independent of the choice of basis $(e_i)$.

For $x \in g$ let $x_h$ be the restriction to $h$ of $\operatorname{ad}_g x$. Then $x \mapsto x_h$ is a representation of $g$ on the vector space $h$ and the restriction $\beta'$ of $\beta$ to $h$ is invariant under this representation. By no. 5, Example 3, the tensor $\sum_{i=1}^n e_i \otimes e'_i$ is independent of the choice of basis $(e_i)$ and is an invariant element of the tensor algebra of $h$. It is also an element of the tensor algebra $T$ of $g$, which is invariant for the representation derived from the adjoint representation of $g$. Its canonical image in $U$, that is $c$, is therefore independent of the choice of basis $(e_i)$ and is an invariant for the representation of $g$ on $U$ considered at the end of no. 2. This element is therefore permutable with every element of $g$ and therefore belongs to the centre of $U$.

When $\beta$ is the bilinear form associated with a $g$-module $M$, the element $c$ of Proposition 11 is called the Casimir element associated with $M$ (or with the corresponding representation). This element exists if the restriction of $\beta$ to $h$ is non-degenerate.