Proposition 10

Let M be an analytic manifold, $\mathfrak{g}$ a complete normable Lie algebra and $\alpha$ an analytic differential form of degree 1 on M, with values in $\mathfrak{g}$, with the following properties: (a) for all $m \in M$, $\alpha_m$ is an isomorphism of $T_m(M)$ onto $\mathfrak{g}$; (b) $d\alpha + [\alpha]^2 = 0$.

Then, for all $m_0 \in M$, there exist an open neighbourhood $M'$ of $m_0$ in M and a Lie group germ structure on $M'$, compatible with the manifold structure on $M'$, with identity element $m_0$ and with the following properties: (i) $\alpha_{m_0}$ is an isomorphism of $L(M')$ onto $\mathfrak{g}$; (ii) the differential form $m \mapsto \alpha_{m_0}^{-1} \circ \alpha_m$ is the left canonical differential form of $M'$. If $M'_1$ and $M'_2$ are two such group germs, $M'_1$ and $M'_2$ have a common open subgroup germ.

There exists a Lie group germ G such that $L(G) = \mathfrak{g}$. Let $m_0 \in M$. By Theorem 5, there exist an open neighbourhood $M'$ of $m_0$ in M and an analytic mapping $f$ of $M'$ into G such that $f(m_0) = e$ and $f^{-1}.df = \alpha$. Then $T_{m_0}(f) = \alpha_{m_0}$ is an isomorphism of $T_{m_0}(M)$ onto $\mathfrak{g}$; hence, shrinking $M'$ and G, it can be assumed that $f$ is an isomorphism of the manifold $M'$ onto the manifold G. We transport to $M'$ the Lie group germ structure on G by means of $f^{-1}$. Then $T_{m_0}(f)$ becomes an isomorphism of $L(M')$ onto $L(G) = \mathfrak{g}$, whence (i). On the other hand, if $\omega$ denotes the left canonical differential form of $G$, then

and hence $m \mapsto \alpha_{m_0}^{-1} \circ \alpha_m$ is the left canonical differential form of $M'$.

Let $M''$ be an open neighbourhood of $m_0$, with a Lie group germ structure, with identity element $m_0$ and with the analogous properties to properties (i) and (ii). Then $\alpha_{m_0}$ is an isomorphism of $L(M')$ onto $g$ and also of $L(M'')$ onto $g$ and hence $L(M') = L(M'')$. Therefore, shrinking $M'$ and $M''$, it can be assumed that there exists an isomorphism $\phi$ of the group germ $M'$ onto the group germ $M''$ (no. 1, Corollary 1 to Theorem 1). Then $\phi^{-1}.d\phi$ is the canonical left differential of $M'$. On the other hand, let $\psi$ be the canonical injection of the manifold $M' \cap M''$ into the Lie group germ $M''$; clearly $\psi^{-1}.d\psi$ is a restriction of the canonical left differential of $M''$. Hence $(\psi^{-1}.d\psi)(m) = \alpha_{m_0}^{-1} \circ \alpha_m = (\phi^{-1}.d\phi)(m)$ for all $m \in M' \cap M''$. Therefore $\phi$ and $\psi$ coincide on a neighbourhood of $m_0$ (§ 3, 18.9). This proves the last assertion of the proposition.