Proposition 5

Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra, $\Phi$ an invariant symmetric bilinear form on $\mathfrak{g}$, and W the Weyl group of $(\mathfrak{g},\mathfrak{h})$. Then the restriction $\Phi '$ of $\Phi$ to $\mathfrak{h}$ is invariant under W. Moreover, if $\Phi$ is non-degenerate, so is $\Phi '$.

Let $\alpha \in R$, let $X_{\alpha}$ be a non-zero element of $\mathfrak{g}^{\alpha},\rho$ the associated representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on $\mathfrak{g}$, and $\pi$ the representation of $\mathbf{S}\mathbf{L}(2, k)$ on $\mathfrak{g}$ compatible with $\rho$. Then $\Phi$ is invariant under $\rho$, and hence under $\pi ($§1, no. 4). In particular, $\Phi '$ is invariant under $\theta_{\alpha}(t)|\mathfrak{h}$ (no. 2), and hence under W. The last assertion follows from Prop. 1 (i).