Lemma 1

Let $\mathfrak{g}$ be a filtered Lie algebra (§ 4, no. 1), $(\mathfrak{g}_\alpha)_{\alpha \in \mathbf{R}}$ its filtration and let $\alpha \in \mathbf{R}$. Let P be a homogeneous Lie polynomial of degree n in the indeterminates $(\Gamma_i)_{i \in I}$ (§ 2, no. 4). Then $P((a_i)) \in \mathfrak{g}_{n\alpha}$ for every family $(a_i)_{i \in I}$ of elements of $\mathfrak{g}_\alpha$.

Every Lie polynomial of degree $n \geqslant 2$ is a finite sum of terms of the form $[Q, R]$ where $Q$ and $R$ are of degree $< n$ and the sum their degrees is equal to $n$ (§ 2, no. 7, Proposition 7). The lemma follows by induction on $n$.

A Lie formal power series† (with coefficients in $K$) in the indeterminates $(T_i)_{i \in I}$ is any element of the Lie algebra $\hat{L}((T_i)_{i \in I}) = \hat{L}(I)$. Such an element $u$ can be written uniquely as the sum of a summable family $(u_\nu)_{\nu \in \mathbf{N}^{(I)}}$ where $u_\nu \in L^\nu(I)$.

Suppose that $I$ is finite. Let $g$ be a complete Hausdorff filtered Lie algebra such that $g = \bigcup_{\alpha > 0} g_\alpha$; let $t = (t_i)_{i \in I}$ be a family of elements of $g$.