Lemma 4

Let H be a compact Lie group, $\rho : H\rightarrow \mathbf{G}\mathbf{L}(V)$ a continuous (hence analytic) representation of H on a finite dimensional real vector space, and W a neighbourhood of the origin in V. There exists an open neighbourhood B of the origin, contained in W and stable under H, and an analytic isomorphism $u: V\rightarrow B$, compatible with the operations of H, such that $u(0) = 0$ and $Du(0) =$ Id$_V$.

Choose a scalar product on V invariant under H (§1, no. 1). There exists a real number $r >0$ such that the open ball B of radius $r$ is contained in W; it is clearly stable under H. Put $u(v) =r(r^2+\|v\|^2)^{-1/2}v$ for all $v\in V$; then $u$ is a bijective analytic map from V to B, compatible with the operations of H, and its inverse map $w \rightarrow r(r^2- \|w\|^2)^{-1/2}w$ is analytic. Moreover, $u(0) = 0$ and $Du(0) =$ Id$_V$.