Proposition 9

Let $\mathfrak{b}$ be the Borel subalgebra of $(\mathfrak{g},\mathfrak{h})$ defined by a chamber C of R. Let $\mathfrak{n}=\mathfrak{g}^{R_+(C)}=\sum_{\alpha\in R,\alpha >0}\mathfrak{g}^{\alpha}$. Let $l=$ dim$\mathfrak{h}$.

(i) If $h\in \mathfrak{h}$ and $x\in \mathfrak{n}$, the characteristic polynomial of ad$_{\mathfrak{g}}(h+x)$ is $T^l\prod_{\alpha\in R}(T-\alpha (h))$.

(ii) The largest nilpotent ideal of $\mathfrak{b}$ is equal to $\mathfrak{n}$ and to $[\mathfrak{b},\mathfrak{b}]$. This is also the set of elements of $\mathfrak{b}$ nilpotent in $\mathfrak{g}$.

(iii) Let B be the basis of R associated to C. For all $\alpha \in B$, let $X_{\alpha}$ be a non-zero element of $\mathfrak{g}^{\alpha}$. Then $(X_{\alpha})_{\alpha\in B}$ generates the Lie algebra $\mathfrak{n}$. We have $[\mathfrak{n},\mathfrak{n}] =\sum_{\alpha\in R,\alpha >0,\alpha \notin B}\mathfrak{g}^{\alpha}$.

There exists a total order on $\mathfrak{h}^*_{\mathbf{Q}}$ compatible with its vector space structure and such that the elements of $R_+(C)$ are $>0$ (Chap. VI, §1, no. 7). Let $h, x$ be as in (i) and $y\in \mathfrak{g}^{\alpha}$. Then $[h+x, y] =\alpha (h)y+z$ where $z\in \sum_{\beta >\alpha}\mathfrak{g}^{\beta}$. Then,

with respect to a suitable basis of $\mathfrak{g}$, the matrix of ad$_{\mathfrak{g}}(h+x)$ has the following properties:

1) it is lower triangular;

2) the diagonal entries of the matrix are the number $0 (l$ times) and the $\alpha (h)$ for $\alpha \in R$.

This proves (i). It also shows that the characteristic polynomial of ad$_{\mathfrak{b}}(h+x)$ is $T\prod_{\alpha\in R_+(C)}^l(T-\alpha (h))$. It follows from the preceding that the

set of elements of $\mathfrak{b}$ nilpotent in $\mathfrak{g}$, as well as the largest nilpotent ideal of $\mathfrak{b}$, are equal to $\mathfrak{n}$. We have $\mathfrak{n}= [\mathfrak{b},\mathfrak{b}]$ by Prop. 5 (i). Finally, assertion (iii) follows from §2, Prop. 4 (ii) and Chap. VI, §1, no. 6, Prop. 19.