Lemma 1

Let $\mathbf{n}\in \mathbf{N}^I$. If $|\mathbf{n}|< r$, then $c^+_r([\mathbf{n}]) = 0$. If $|\mathbf{n}|=r$, then

where $\varphi$ belongs to the set of maps from $\{1,2, . . . , r\}$ to I which take each value $i\in In_i$ times.

By Prop. 1,

where the summation extends over the set of sequences $(\mathbf{p}_1, . . . ,\mathbf{p}_r)$ of $r$ elements of M such that $\mathbf{p}_1+\cdots +\mathbf{p}_r=\mathbf{n}$. In view of Chap. II, §1, no. 3, Prop. 6, the map $c^+_r$, extended by linearity to a map from $U(\mathfrak{g})$ to $\mathbf{T}^r(U^+(\mathfrak{g}))$, vanishes on $U_{r-1}(\mathfrak{g})$. It follows that, for $r\geq |\mathbf{n}|$,

For $r >|\mathbf{n}|$, the relation $\mathbf{p}_1+\cdots +\mathbf{p}_r=\mathbf{n}$ implies that at least one of the $\mathbf{p}_i$ is zero, so $c^+_r([\mathbf{n}]) = 0$. For $r=|\mathbf{n}|$, the only non-zero terms of the third member of (8) are those for which $|\mathbf{p}_1|=\cdots =|\mathbf{p}_r|= 1$, hence (7).

We return to the proof of Th. 1. We retain the notations of (*) and prove, by descending induction on $|\mathbf{n}|$, that $d$ divides $a_{\mathbf{n}}$, which is clear when $|\mathbf{n}|$ is sufficiently large. If $d$ divides $a_{\mathbf{n}}$ for $|\mathbf{n}|> r$ then, putting

we have

For any map $\varphi$ from $\{1,2, . . . , r\}$ to I, put

and $a_{\varphi}=a_{\mathbf{n}}$ where $\mathbf{n}=$ (Card$\varphi^{-1}(i)$)$_{i\in I}$. By Lemma 1, (9) implies that

so $c^+_r(u')\in \mathbf{T}^r(\mathscr{U}^+)\cap \mathbf{Q}\mathbf{T}^r(\mathscr{G})$. But the submodule $\mathscr{G}$ of $\mathscr{U}^+$ is a direct factor (Algebra, Chap. VII, §4, no. 3, Cor. of Th. 1), so the submodule $\mathbf{T}^r(\mathscr{G})$ is a direct factor of $\mathbf{T}^r(\mathscr{U}^+)$, and hence $c^+_r(u')\in \mathbf{T}^r(\mathscr{G})$. On the other hand, the $x_i$ form a basis of $\mathscr{G}$ by hypothesis, so the $e_{\varphi}$ form a basis of $\mathbf{T}^r(\mathscr{G})$. Then (10) proves that $d$ divides the $a_{\varphi}$, that is, the $a_{\mathbf{n}}$ for $|\mathbf{n}|=r$. This proves (*).