Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional inner product space $W$. Moreover we have: $$\forall \xi \in \mathfrak{g}, \forall u,v\in W\qquad <\rho(\xi).u,v>=-<\rho(\xi).v),u>.$$ For any arbitrary basis $\{\xi_i\}_{i=1}^{n}$ of $\mathfrak{g}$, define a linear map $H:W\rightarrow W$, with $H(u)=\sum_{i=1}^{n}\rho(\xi_i).(\rho(\xi_i).u))$. Then $H$ is a well-defined symmetric linear transformation on $W$.

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What can we say about eigenvalues of $H$ and their relation with the properties of representation $\rho$? Are there some geometric interpretations for the eigenvalues?