Corollary

Let $M$ be an analytic manifold of finite dimension $n$. Let $\omega_1, \ldots, \omega_n$ be analytic differential forms of degree 1 on $M$, with scalar values, which are linearly independent at each point of $M$ and such that, for all $k = 1, \ldots, n$, $d\omega_k$ is a linear combination with constant coefficients of the $\omega_i \wedge \omega_j$. Then, for all $m_0 \in M$, there exists 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 such that $\omega_1|_{M'}, \ldots, \omega_n|_{M'}$ form a basis of the space of left invariant differential forms on $M'$ of degree 1 with scalar values.

If $M'_1$ and $M'_2$ are two such group germs, $M'_1$ and $M'_2$ have a common open subgroup germ.

Let $X_1, \ldots, X_n$ be the vector fields on $M$ such that, at each point $m$ of $M$, the $(X_t)_m$ constitute the basis of $T_m(M)$ dual to $((\omega_1)_m, \ldots, (\omega_n)_m)$. These fields are analytic. By hypothesis, there exist $c_{ijk} \in \mathbf{K}$ ($1 \leq i, j, k \leq n$) such that $c_{ijk} = -c_{jik}$ and $d\omega_k = \sum_{i < j} c_{ijk} \omega_i \wedge \omega_j$. By Differentiable and Analytic Manifolds, R, 8.5.7, formula (11),

and hence $[X_i, X_j] = -\sum_k c_{ijk} X_k$. It then follows that the $-c_{ijk}$ are the constants of structure of a Lie algebra $g$ relative to a basis $(e_1, \ldots, e_n)$. For all $m \in M$, let $\alpha_m$ be the linear mapping of $T_m(M)$ into $g$ which maps $(X_1)_m$ to $e_1, \ldots, (X_n)_m$ to $e_n$. Then $\alpha$ is an analytic differential form on $M$ of degree 1 with values in $g$ and $\alpha_m$ is an isomorphism of $T_m(M)$ onto $g$. On the other hand, $\alpha = \sum_{k=1}^n \omega_k e_k$ and hence and It then suffices to apply Proposition 10.