Lemma 7

The ideal $\mathfrak{n}$ of $\mathfrak{a}_+$ generated by the $x_{\alpha \beta}(\alpha , \beta \in B, \alpha \not=\beta )$ is an ideal of $\mathfrak{a}$. The ideal of $\mathfrak{a}_-$ generated by the $y_{\alpha \beta}(\alpha , \beta \in B, \alpha \not=\beta )$ is an ideal of $\mathfrak{a}$ and is equal to $\theta (\mathfrak{n})$.

Let $\mathfrak{n}'=\sum_{\alpha ,\beta\in B,\alpha\not=\beta}kx_{\alpha \beta}$. Since each $x_{\alpha \beta}$ is homogeneous in $\mathfrak{a}, [\mathfrak{a}^0,\mathfrak{n}']\subset \mathfrak{n}'$

(Lemma 5 and Prop. 2). Let U (resp. V) be the enveloping algebra of $\mathfrak{a}$ (resp. $\mathfrak{a}_+$), and $\sigma$ the representation of U on $\mathfrak{a}$ defined by the adjoint representation of $\mathfrak{a}$. The ideal of $\mathfrak{a}$ generated by $\mathfrak{n}'$ is $\sigma (U)\mathfrak{n}'$. Now $\mathfrak{a}=\mathfrak{a}_++\mathfrak{a}^0+\mathfrak{a}_-$ (Prop. $2$)$,\sigma (\mathfrak{a}_-)\mathfrak{n}'= 0$ (Lemma 6), and $\sigma (\mathfrak{a}^0)\mathfrak{n}'\subset \mathfrak{n}'$ by the preceding. By the Poincaré-Birkhoff-Witt theorem, $\sigma (U)\mathfrak{n}'=\sigma (V)\mathfrak{n}'$, which proves the first assertion of the lemma. It follows that the ideal of $\theta (\mathfrak{a}_+) =\mathfrak{a}_-$ generated by the $\theta (x_{\alpha \beta}) =y_{\alpha \beta}(\alpha , \beta \in B, \alpha \not=\beta )$ is the ideal $\theta (\mathfrak{n})$ of $\mathfrak{a}$. Q.E.D.

The ideal $\mathfrak{n}+\theta (\mathfrak{n})$ of $\mathfrak{a}$ is graded since it is generated by homogeneous elements. Consequently, the Lie algebra $\mathfrak{a}/(\mathfrak{n}+\theta (\mathfrak{n}))$ is a Q-graded Lie algebra; in the remainder of this paragraph, it is denoted by $\mathfrak{g}_B$, or simply by $\mathfrak{g}$. By Prop. 2, if $\mathfrak{g}^\mu\not= 0$ then $\mu\in Q_+$, or $\mu\in Q_-$, or $\mu= 0$. Denote by $X_{\alpha}$ (resp. $H_{\alpha}, X_{-\alpha}$) the canonical image of $x_{\alpha}$ (resp. $h_{\alpha}, x_{-\alpha}$) in $\mathfrak{g}$. In view of the definition of $\mathfrak{a},\mathfrak{n}$ and $\theta (\mathfrak{n})$, it follows that $\mathfrak{g}$ is the Lie algebra defined by the generating family $((X_{\alpha}, H_{\alpha}, X_{-\alpha}))_{\alpha\in B}$ and the relations

(ad $X_{\alpha}$)$^{1-n(\beta ,\alpha)}X_{\beta}= 0(\alpha \not=\beta )$ (21)

(ad $X_{-\alpha}$)$^{1-n(\beta ,\alpha)}X_{-\beta}= 0(\alpha \not=\beta )$. (22)

Let $z\in \mathfrak{g}$ and $\mu\in Q$. Then $z\in \mathfrak{g}^\mu$ if and only if $[H_{\alpha}, z] =\langle \mu, \alpha^{\vee}\rangle z$ for all $\alpha \in B$. This follows from Lemma 5.

Since $\mathfrak{a}^0\cap (\mathfrak{n}+\theta (\mathfrak{n})) = 0$, the canonical map from $\mathfrak{a}^0$ to $\mathfrak{g}^0$ is an isomorphism. Consequently, $(H_{\alpha})_{\alpha\in B}$ is a basis of the vector space $\mathfrak{g}^0$. The commutative subalgebra $\mathfrak{g}^0$ of $\mathfrak{g}_B$ will be denoted by $\mathfrak{h}_B$ or simply by $\mathfrak{h}$. There exists a unique isomorphism $\mu \rightarrow \mu_B$ from V to $\mathfrak{h}^*$ such that $\langle \mu_B, H_{\alpha}\rangle =\langle \mu, \alpha^{\vee}\rangle$ for all $\mu\in V$ and all $\alpha \in B$.

The involutive automorphism $\theta$ of $\mathfrak{a}$ defines by passage to the quotient an involutive automorphism of $\mathfrak{g}$ that will also be denoted by $\theta$. We have $\theta (X_{\alpha}) =X_{-\alpha}$ for $\alpha \in B\cup (-B)$, and $\theta (H_{\alpha}) =-H_{\alpha}$.