Proposition 2

Let

If $(x, y, z) \in \Omega'$, then

and

Let $(x, y, z) \in \Omega'$; clearly $(x, y) \in \Omega$ and $(y, z) \in \Omega$. Moreover:

and hence by (13)

Now $M(\|x\| + \|y\|) < \log \frac{3}{2} - M\|z\|$; we write $u = \exp(M\|z\|)$; then $1 \leq u \leq \frac{3}{2}$ and

We see similarly that $(x, h(y, z)) \in \Omega$.

We now prove (17). In the Lie algebra $\hat{L}(\{U, V, W\})$,

by Proposition 4 of § 6, no. 5. By no. 1, formula (2), we therefore have in $\hat{P}(g \times g \times g, g)$ the relation

By Differentiable and Analytic Manifolds, R, 3.1.9, there exists a number $\varepsilon > 0$ such that formula (17) is true when $\|x\|, \|y\|$ and $\|z\|$ are $\leq \varepsilon$. But the functions $(x, y, z) \mapsto h(h(x, y), z)$ and $(x, y, z) \mapsto h(x, h(y, z))$ are analytic functions on $\Omega'$ with values in $g$ (Differentiable and Analytic Manifolds, R, 3.2.7). As $\Omega'$ is connected and they coincide in a neighbourhood of 0, they are equal (Differentiable and Analytic Manifolds, R, 3.2.5).

The above results imply:

Let $\alpha$ be a real number such that $0 < \alpha \leq \frac{1}{3M} \log \frac{3}{2}$. Let

and $m : \Theta \to G$ be the restriction of $h$ to $\Theta$. Then: (1) $\Theta$ is open in $G \times G$ and $m$ is analytic. (2) $x \in G$ implies $(0, x) \in \Theta$, $(x, 0) \in \Theta$ and $m(0, x) = m(x, 0) = x$. (3) $x \in G$ implies $-x \in G$, $(x, -x) \in \Theta$, $(-x, x) \in \Theta$ and

(4) Let x, y, z be elements of G such that (x, y) ∈ Θ, (m(x, y), z) ∈ Θ, (y, z) ∈ Θ and (x, m(y, z)) ∈ Θ. Then m(m(x, y), z) = m(x, m(y, z)).

In other words (Chapter III, § 1), if we write −x = σ(x), the quadruple (G, 0, σ, m) is a Lie group germ over K.