Proposition 3

If $a \in m, b \in m$, then $\exp H(a, b) = \exp a . \exp b$.

Let $a, b$ be in $m$; there exists $\alpha > 0$ such that $a \in A_\alpha$ and $b \in A_\alpha$. Then there exists a continuous homomorphism $\theta$ of the Magnus algebra $\hat{A}(\{U, V\})$ into $A$ mapping $U$ to $a$ and $V$ to $b$ (§ 5, no. 1, Proposition 1).

The restriction of $\theta$ to $\hat{L}(\{U, V\})$ is a continuous homomorphism of Lie algebras of $L(\{U, V\})$ into $m$ which maps $U$ (resp. $V$) to $a$ (resp. $b$). By formula (6) of no. 3, therefore $\theta(H) = H(a, b)$. It then suffices to apply the continuous homomorphism $\theta$ to the two sides of the relation

taking account of Remark 3 of no. 1.