Proposition 3

The Lie algebra $\mathfrak{a}_+$ (resp. $\mathfrak{a}_-$) is a free Lie algebra with basic family $(x_{\alpha})_{\alpha\in B}$ (resp. $(x_{-\alpha})_{\alpha\in B}$) (cf. Chap. II, §2, no. 3).

Let L be the Lie subalgebra of E generated by B. By Chap. II, §3, Th. 1, L can be identified with the free Lie algebra generated by B. The left regular representation of E on itself is clearly injective, and defines by restriction to L an injective representation $\rho '$ of the Lie algebra L on E. Let $\varphi$ be the unique homomorphism from L to $\mathfrak{a}_-$ which takes $\alpha$ to $x_{-\alpha}$ for all $\alpha \in B$. Then, for all $\alpha \in B,\rho (\varphi (\alpha ))$ is the endomorphism of left multiplication by $\alpha$ on E, so $\rho \circ \varphi =\rho '$, which proves that $\varphi$ is injective. Thus, $(x_{-\alpha})_{\alpha\in B}$ is a basic family for $\mathfrak{a}_-$. Since $\theta (x_{-\alpha}) =x_{\alpha}$ for all $\alpha$ (cf. Lemma $4$)$, (x_{\alpha})_{\alpha\in B}$ is a basic family for $\mathfrak{a}_+$.