diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -690,27 +690,23 @@ known as \emph{the Casimir element of a \(\mathfrak{g}\)-module}.
\begin{proof}
To see that \(\Omega_M\) is central fix a basis \(\{X_i\}_i\) for
- \(\mathfrak{g}\) and denote by \(\{X^i\}_i\) its dual basis as in
- Definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
- by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
- in \([X, X_i]\) and \([X, X^i]\), respectively.
-
- The invariance of \(\kappa_M\) implies
- \[
- \lambda_{i k}
- = \kappa_M([X, X_i], X^k)
- = \kappa_M(-[X_i, X], X^k)
- = \kappa_M(X_i, -[X, X^k])
- = - \mu_{k i}
- \]
-
- Hence
+ \(\mathfrak{g}\) and denote by \(\{X^i\}_i\) its dual basis with respect to
+ \(\kappa_M\), as in
+ Definition~\ref{def:casimir-element}. Given any \(X \in \mathfrak{g}\), it
+ follows from definition of the \(X^i\) that \(X = \kappa_M(X, X^1) X_1 +
+ \cdots + \kappa_M(X, X^r) X_r = \kappa_M(X, X_1) X^1 + \cdots + \kappa_M(X,
+ X_r) X^r\).
+
+ In particular, it follows from the invariance of \(\kappa_M\) that
\[
\begin{split}
[X, \Omega_M]
& = \sum_i [X, X_i X^i] \\
& = \sum_i [X, X_i] X^i + \sum_i X_i [X, X^i] \\
- & = \sum_{i j} \lambda_{i j} X_j X^i + \sum_{i j} \mu_{i j} X_i X^j \\
+ & = \sum_{i j} \kappa_M([X, X_i], X^j) X_j X^i
+ + \sum_{i j} \kappa_M([X, X^i], X_j) X_i X^j \\
+ & = \sum_{i j} (\kappa_M([X, X_j], X^i) + \kappa_M(X_j, [X, X^i]))
+ X_i X^j \\
& = 0
\end{split},
\]