Proposition 4

(i) *Let $X$ be a subset of $\mathbf{R}$, let $V_X$ be the vector subspace of $V$ generated by $X$, and let $V'_X$ be the vector subspace of $V^*$ generated by the $\alpha^-$, where $\alpha \in X$. Then $V$ is the direct sum of $V_X$ and the orthogonal complement of $V'_X$; $V^*$ is the direct sum of $V'_X$ and the orthogonal complement of $V_X$, and $V'_X$ is identified with the dual of $V_X$.

(ii) $R \cap V_X$ is a root system in $V_X$, and the canonical bijection from $R \cap V_X$ to its inverse root system is identified with the restriction of the map $\alpha \mapsto \alpha^-$ to $R \cap V_X$.

By Remark 2, we can assume that $k = \mathbf{R}$. Identify $V$ with $V^*$ by means of the symmetric bilinear form of Prop. 3. We have $\alpha^- = \frac{2\alpha}{(\alpha|\alpha)}$ for all $\alpha \in R$ (Lemma 2). Every vector subspace of $V$ is non-isotropic, and the proposition is now clear.