Proposition 3

(i) For all $\alpha \in \mathbf{R}$, there exists a unique group homomorphism $g_\alpha : \mathrm{gr}_\alpha(G) \to \mathrm{gr}_\alpha(A)$ which maps the class modulo $G_\alpha^+$ of an element $a \in G_\alpha$ to the class modulo $A_\alpha^+$ of $\rho(a) - 1$.

(ii) Let g be the group homomorphism of gr(G) into gr(A) whose restriction to gr_\alpha(G) is g_\alpha for all \alpha. The mapping g is an injective homomorphism of Lie \mathbf{Z}-algebras.

(i) Let \alpha > 0. By hypothesis, for all a in G_\alpha, \rho(a) - 1 \in A_\alpha; let p_\alpha(a) denote the class of \rho(a) - 1 modulo A_\alpha^+. As A_{2\alpha} \subset A_\alpha^+, relation (25) implies p_\alpha(ab) = p_\alpha(a) + p_\alpha(b). Then a \in G_\alpha^+ if and only if \rho(a) - 1 \in A_\alpha^+; therefore G_\alpha^+ is the kernel of the homomorphism p_\alpha of G_\alpha into gr_\alpha(A). On passing to the quotient, p_\alpha then defines an injective homomorphism g_\alpha of gr_\alpha(G) into gr_\alpha(A).

For \alpha \leq 0, gr_\alpha(G) = \{0\} and the only choice is g_\alpha = 0.

(ii) As g_\alpha is injective for all real \alpha, g is injective. We show that g is a Lie algebra homomorphism. As gr_\alpha(G) = \{0\} for \alpha \leq 0, it suffices to establish the formula

for \alpha > 0, \beta > 0, \rightarrow \in G_\alpha and b \in G_\beta, which follows from (26).