Let $g$ be a simple Lie algebra over $\mathbb{C}$. Let $\Omega$ be the Casimir element of $g \otimes g$ associated to a non-degenerated invariant form on $g$. How to show that $( \Lambda^3 g)^{g} = \mathbb{C}Z$? Here $Z = [\Omega_{12}, \Omega_{23}]$. Any help will be greatly appreciated!
Edit: this result is on page 41, Theorem 5.2 of the book.