Exercise 8, § 2

a) Let $\mathfrak{g}$ be a Lie algebra and $n$ an integer $\geq 0$. Let $\Sigma_n(\mathfrak{g})$ be the subset of $\mathfrak{g}^n$ consisting of the families $(x_1, . . . , x_n)$ that generate $\mathfrak{g}$ as a $k$-algebra. Show that $\Sigma_n(\mathfrak{g})$ is open in $\mathfrak{g}^n$ in the Zariski topology (Chap. VII, App. I). If $k'$ is an extension of $k$, then $\Sigma_n(\mathfrak{g}_{(k')})\cap \mathfrak{g}^n=\Sigma_n(\mathfrak{g})$. Deduce that, if $\mathfrak{g}_{(k')}$ can be generated by $n$ elements, so can $\mathfrak{g}$.

b) Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra. Let $x$ be an element of $\mathfrak{h}$ such that $\alpha (x)\not= 0$ for all $\alpha \in R$ and $\alpha (x)\not=\beta (x)$ for any pair of distinct elements $\alpha , \beta \in R$. For all $\alpha \in R$, let $y_{\alpha}$ be a non-zero element of $\mathfrak{g}^{\alpha}$, and let $y=\sum_{\alpha\in R}y_{\alpha}$. Show that, for all $\alpha \in R$, there exists a polynomial $P_{\alpha}(T)\in k[T]$, without constant term, such that $y_{\alpha}= P_{\alpha}$(ad $x$)$.y$. Deduce that $\mathfrak{g}$ is generated by $\{x, y\}$.

c) Show, by means of a) and b), that every semi-simple Lie algebra can be generated by two elements.