Proposition 12

*Let $g$ be a Lie algebra over a field $K$, $h$ an ideal of $g$ of finite dimension n and M a g-module of finite dimension over K. Let c be the Casimir element (assumed to exist) associated with M and h.

(a) $\operatorname{Tr}(c_M) = n$. (b) If M is simple and n is not divisible by the characteristic of K, $c_M$ is an automorphism of M.

In the notation of Proposition 11,

Hence, if n is not divisible by the characteristic of K, $c_M \neq 0$. On the other hand, as c belongs to the centre of U, $c_M$ is permutable with all the $x_M, x \in g$. If further M is simple, $c_M$ is therefore invertible in $\mathscr{L}(M)$ (Algebra, Chapter VIII, § 4, no. 3, Proposition 2).