Proposition 1

(i) Let $\alpha, \beta$ be in $\mathbf{R}$. There exists a biadditive mapping

which maps $(xG_\alpha^+, yG_\beta^+)$ onto $(x, y)G_{\alpha+\beta}^+$.

(ii) Let $\phi$ be the biadditive mapping of $\mathrm{gr}(G) \times \mathrm{gr}(G)$ into $\mathrm{gr}(G)$ whose restriction to $\mathrm{gr}_\alpha(G) \times \mathrm{gr}_\beta(G)$ is $\phi_{\alpha\beta}$ for every ordered pair $(\alpha, \beta)$. The mapping $\phi$ gives $\mathrm{gr}(G)$ a Lie $\mathbf{Z}$-algebra structure.

(i) Recall the identity

for $x, x', y$ in $G$ (Algebra, Chapter I, § 6, no. 2, formula (4 bis)).

For $x \in G_\alpha$ and $y \in G_\beta$, the class modulo $G_{\alpha+\beta}^+$ of the element $(x, y)$ of $G_{\alpha+\beta}$ will be denoted by $f(x, y)$. For $a$ in $G_{\alpha+\beta}$ and $x'$ in $G$, $a^{-1}.ax' = (a, x') \in G_{\alpha+\beta}^+$; in particular $f(x, y)$ is equal to the class modulo $G_{\alpha+\beta}^+$ of $(x, y)x'$. Formula (15) therefore implies

Now $(y, x) = (x, y)^{-1}$, whence

From (16) and (17) we deduce

We have to prove that the mapping $f : G_\alpha \times G_\beta \to \mathrm{gr}_{\alpha+\beta}(G)$ defines on taking quotients a mapping $\phi_{\alpha\beta} : \mathrm{gr}_\alpha(G) \times \mathrm{gr}_\beta(G) \to \mathrm{gr}_{\alpha+\beta}(G)$. By (16) and (18) it suffices to prove that $f(x, y) = 0$ if $x \in G_\alpha^+$ or $y \in G_\beta^+$, which follows from (13').

(ii) As $(x, x) = e$, it follows from (17) that $\phi$ is an alternating $\mathbf{Z}$-bilinear mapping. Hence it remains to prove that, for $u \in \mathrm{gr}_\alpha(G)$, $v \in \mathrm{gr}_\beta(G)$ and $w \in \mathrm{gr}_\gamma(G)$,

Let $x \in G_\alpha, y \in G_\beta$ and $z \in G_\gamma$ be elements representing respectively $u, v$ and $w$. We know that $x^y$ and $x$ are two elements of $G_\alpha$ which are congruent modulo $G_\alpha^+$ and hence $x^y$ is a representative of $u$ in $G_\alpha$; as $(y, z)$ is a representative of $\phi(v, w)$ in $G_{\beta+\gamma}$, we see that $(x^y, (y, z))$ is a representative of $\phi(u, \phi(v, w))$ in $G_{\alpha+\beta+\gamma}$. By cyclic permutation, we see that $(y^z, (z, x))$ and $(z^x, (x, y))$ represent respectively $\phi(v, \phi(w, u))$ and $\phi(w, \phi(u, v))$ in $G_{\alpha+\beta+\gamma}$. Relation (19) is then a consequence of the following identity (Algebra, Chapter I, § 6, no. 2, formula (15)):

The Lie algebra $\mathrm{gr}(G)$ over $\mathbf{Z}$ defined in Proposition 1 is called the graded Lie algebra associated with the filtered group $G$.