Proposition 1

Let $\alpha ,\beta$ be roots of $(\mathfrak{g},\mathfrak{h})$ and let $\langle \cdot ,\cdot \rangle$ be a non-degenerate invariant symmetric bilinear form on $\mathfrak{g}($for example the Killing form of $\mathfrak{g})$.

(i) If $\alpha +\beta \not= 0,\mathfrak{g}^{\alpha}$ and $\mathfrak{g}^{\beta}$ are orthogonal. The restriction of $\langle \cdot ,\cdot \rangle$ to $\mathfrak{g}^{\alpha}\times \mathfrak{g}^{-\alpha}$ is non-degenerate. The restriction of $\langle \cdot ,\cdot \rangle$ to $\mathfrak{h}$ is non-degenerate.

(ii) Let $x\in \mathfrak{g}^{\alpha},y\in \mathfrak{g}^{-\alpha}$ and $h\in \mathfrak{h}$. Then $[x, y]\in \mathfrak{h}$ and

Assertion (i) is a particular case of Prop. 10 (iii) of Chap. VII, §1, no. 3. If $x\in \mathfrak{g}^{\alpha},y\in \mathfrak{g}^{-\alpha}$ and $h\in \mathfrak{h}$, we have $[x, y]\in \mathfrak{g}^{\alpha-\alpha}=\mathfrak{h}$, and