Example 2

Let M be a K-module with a finite basis. If M has a g-module structure, $\mathscr{L}(M, M)$ and $M^* \otimes M$ have g-module structures and the canonical mapping of $M^* \otimes M$ into $\mathscr{L}(M, M)$ is a g-module isomorphism (Proposition 4). As $1 \in \mathscr{L}(M, M)$ is obviously an invariant (cf. Example 1), the corresponding element $u$ of $M^* \otimes M$ is an invariant. If $(e_i)_{1 \leq i \leq n}$ is a basis of M and $(e_i^*)_{1 \leq i \leq n}$ is the dual basis, we have $u = \sum_{i=1}^n e_i^* \otimes e_i$.