Corollary

Let $\rho$ be a finite dimensional representation of $\mathfrak{g}$ and $\Phi$ the bilinear form associated to $\rho$.

(i) If $x, y\in \mathfrak{h}_{\mathbf{Q}}$, then $\Phi (x, y)\in \mathbf{Q}$ and $\Phi (x, x)\in \mathbf{Q}_+$.

(ii) If $\rho$ is injective, the restriction of $\Phi$ to $\mathfrak{h}$ is non-degenerate.

Assertion (i) follows from Prop. 1 since the elements of P have rational values on $\mathfrak{h}_{\mathbf{Q}}$. If $\rho$ is injective, $\Phi$ is non-degenerate (Chap. I, §6, no. 1, Prop. 1), so the restriction of $\Phi$ to $\mathfrak{h}$ is non-degenerate (Chap. VII, §1, no. 3, Prop. 10 (iii)).