Let $L$ be a finite extension of $\mathbb{Q}\_p$, $G$ be a locally $L-$analytic group, and let $H$ be a compact open subgroup of $G$. Then in corollary $5.3.19$ of [this paper](http://www.math.uchicago.edu/~emerton/pdffiles/analytic.pdf) Emerton shows that the space of locally analytic distributions $\mathcal{D}^{la}(H,K)$ is a Frechet-Stein algebra. In the beginning of the proof, the author says that it suffices to prove the case when $H$ is the $L-$valued point of a **good** analytic open subgroup which is obtained by exponentiating a sufficiently small Lie algebra $\mathfrak{h}$ with the property that $[\mathfrak{h},\mathfrak{h}] \subset a\mathfrak{h}$ where $a^{p-1} \in p\mathcal{O}\_L$. This is what I have some trouble understanding. Explanations are welcome.



Locally analytic distribution algebra is a Frechet-Stein algebra
The basic idea is that any compact open $H$ contains good open subgroup $H'$. (This is essentially just by taking any sufficiently small neighborhood of the origin in the Lie algebra and applying the exponential map. If $\mathfrak{h}$ is a $\mathbb{Z}\_p$-lattice in the Lie algebra of $H$ and $[\mathfrak{h},\mathfrak{h}] \subset p^{-n} \mathfrak{h}$ for some $n$, then $[p^{n+1} \mathfrak{h},p^{n+1} \mathfrak{h}] \subset p(p^{n+1} \mathfrak{h})$.) Since $H$ is compact, $H'$ has finite index in $H$. The distribution algebra $D(H,K)$ is finite free as a $D(H',K)$-module, and so $D(H,K)$ inherits a Frechet-Stein structure from $D(H',K)$.


As Emerton mentions, more details can be found in the following references:


* Serre, *Lie algebras and Lie groups*, LG IV.8 for the existence of good subgroups
* Schneider and Teitelbaum, *Algebras of p-adic distributions and admissible representations*, Theorem 5.1 for the proof that if $D(H',K)$ Frechet-Stein, then $D(H,K)$ is as well


