Proposition 52

Let n be an integer > 0, G an n-dimensional Lie group, U a symmetric open neighbourhood of e in G and $\psi : U^2 \to K^n$ a chart of G such that $\psi(e) = 0$. If $(x_1, \ldots, x_n)$ are the coordinates of $x \in \psi(U)$ and $(y_1, \ldots, y_n)$ the coordinates of $y \in \psi(U)$, we denote by

the coordinates of $\psi(\psi^{-1}(x)^{-1}\psi^{-1}(y))$. Then, if we write, for $1 \leq k \leq n$,

the differential forms $\varpi_k$ on $\psi(U)$ are derived through $\psi$ from left invariant differential forms on G and are such that $\varpi_k(0, \ldots, 0) = dx_k$.

We apply Lemma 2 with $E = K$, taking $\phi(g)$ to be the coordinate of $\psi(g)$ of index k. We obtain a differential form $\omega_k$; let $\varpi_k$ be its transform under $\psi$. The value of $\varpi_k$ at $(x_1, \ldots, x_n)$ is the differential at $(x_1, \ldots, x_n)$ of the function $y \mapsto m_k(x_1, \ldots, x_n, y_1, \ldots, y_n)$; this value is therefore given by formula (32). It then suffices to use the conclusion to Lemma 2.