Proposition 4

Let $a\in G$ and H be a Lie subgroup germ of G with Lie algebra $\mathfrak{g}^1(a)$. The map $(b, c) \rightarrow cabc^{-1}$ from $H\times G$ to G is a submersion at $(e, e)$.

Indeed, let $\mathfrak{m}$ be a complement of $\mathfrak{g}^1(a)$ in $\mathfrak{g}$ and exp an exponential map of G defined on an open neighbourhood U of 0 in $\mathfrak{g}$. We can choose U so that exp(U $\cap \mathfrak{g}^1(a)$)$\subset H$. The map $f: (x, y) \rightarrow$ (exp $x$, exp $y$) is an analytic map on a neighbourhood of $(0,0)$ in $\mathfrak{g}^1(a)\times \mathfrak{m}$ with values in $H\times G$. By Lemma 3, the composite of $f$ with the map $\varphi : (b, c) \rightarrow cabc^{-1}$ is étale at $(0,0)$. It follows that $\varphi$ is a submersion at $f(0,0) = (e, e)$.