Exercise 14, § 3

Retain the notations of Prop. 9.

a) Let $\alpha \in B$. Show that $\mathfrak{n}\cap$ Ker ad $X_{\alpha}$ is the direct sum of the $\mathfrak{g}^{\beta}$, where $\beta$ belongs to the set of elements of $R_+$ such that $\alpha +\beta \notin R_+$.

b) Deduce that, if $\mathfrak{g}$ is simple, the centre of $\mathfrak{n}$ is equal to $\mathfrak{g}^{\widetilde{\alpha}}$, where $\widetilde{\alpha}$ is the highest root of $R_+$. In the general case, the dimension of the centre of $\mathfrak{n}$ is equal to the number of simple components of $\mathfrak{g}$.