Exercise 2, § 4

Retain the notations of no. 3. Let $\mathfrak{m}$ be an ideal of $\mathfrak{a}$.

a) Let $\alpha$ and $\beta$ be two distinct elements of B. Assume that (ad $x_{\alpha}$)$^Nx_{\beta}$ belongs to $\mathfrak{m}$ for N sufficiently large. Show that $x_{\alpha \beta}\in \mathfrak{m}$. (Apply the results of §1, no. 2 to $\mathfrak{a}/\mathfrak{m}$, provided with a suitable $\mathfrak{s}\mathfrak{l}(2, k$)-module structure.)

b) Show that $\mathfrak{n}+\theta \mathfrak{n}$ is the smallest finite codimensional ideal of $\mathfrak{a}$. Show that this is also the smallest ideal containing the (ad $x_{\alpha}$)$^4x_{\beta}$ and the (ad $x_{-\alpha}$)$^4x_{-\beta}$.