Remark

(2) Take $g$ to be the Lie algebra $\hat{L}(X)$. The group law introduced in the above proposition coincides with the law defined in no. 2. In other words,

thus the Hausdorff group law is given by the Hausdorff series.

(3) Let $g$ be a Lie algebra with the integral filtration ($\mathscr{C}^n g$) defined by the lower central series. Suppose that there exists $m \geq 1$ such that $\mathscr{C}^m g = \{0\}$. With the topology derived from the filtration $(\mathscr{C}^n g)_{n \geq 1}$, the Lie algebra $g$ is Hausdorff, complete and even discrete. Then $P(a_1, \ldots, a_r) = 0$ for $a_1, \ldots, a_r$ in $g$ and for every homogeneous Lie polynomial $P$ of degree $\geq m$; in particular, $H_{r,s}(a, b) = 0$ for $r + s \geq m$ and the series $H(a, b) = \sum_{r,s} H_{r,s}(a, b)$ has only a finite number of non-zero terms. The group law $(a, b) \mapsto H(a, b)$ on $g$ is then a polynomial mapping (§ 2, no. 4).