Theorem 2

If r and s are two positive integers such that $r + s \geq 1$, then $H_{r,s} = H'_{r,s} + H''_{r,s}$, where

In $\hat{A}_\mathbf{Q}(\{U, V\})$, $\exp U \cdot \exp V = 1 + W$, where $W = \sum_{r+s \geq 1} \frac{U^r V^s}{r! s!}$, whence

The linear mapping $P_n$, defined by $P_n(x_1, \ldots, x_n) = \frac{1}{n} \left( \prod_{i=1}^{n-1} (\mathrm{ad}\, x_i) \right)(x_n)$ for $n \geq 1$ and $x_1, \ldots, x_n$ in $\{U, V\}$, is a projector of $A^n_\mathbf{Q}(\{U, V\})$ onto $L^n_\mathbf{Q}(\{U, V\})$ (§ 3, no. 2, Corollary to Proposition 1); as $H_{r,s}$ belongs to $L^{r+s}_\mathbf{Q}(\{U, V\})$, $H_{r,s} = P_{r+s}(H_{r,s})$. Now

when $s_m \geqslant 1$ and

when $r_m \geqslant 1$ and $s_m = 0$. Moreover, obviously $(\mathrm{ad}\ t)^{p-1}.t = 0$ if $p \geqslant 2$ and $(\mathrm{ad}\ t)^0.t = t$. It follows that the two sides of (12) are zero when $s_m \geqslant 2$ and those of (13) are zero when $r_m \geqslant 2$. The theorem then follows since $H'_{r,s}$ is the sum of the terms of type (12) and $H''_{r,s}$ is the sum of the terms of type (13).