Decomposing maximal compact subgroups of SO(n,1)

Question

Let $G=SO(n,1)$ and let $G=KAN$ be an Iwasawa decomposition of $G$. Let $M$ be the centralizer of $A$ in $K$. In this case, we have $K≃SO(n)$, $A≃\Bbb R$(this is the maximal diagonalizable subgroup), $N≃\Bbb{R}^{n−1}$ and $M≃SO(n−1)$. Let $\mathfrak k$ be the Lie algebra of $K$ and $\mathfrak m\subseteq\mathfrak k$ be the lie algebra of $M$. Let $\mathfrak h$ be the orthocomplement to $\mathfrak m$ in $\mathfrak k$. That is $\mathfrak k= \mathfrak m\oplus \mathfrak h$. Let $H$ be the Lie hgroup associated to generate by $exp(\mathfrak h)$. What can we say about $HM$? When is it true that $HM=K$?

I believe $K/HM$ is discrete and finite. In this case there exists $e=\omega_1,\omega_2,\dots,\omega_l$ s.t. $K=HM\sqcup\omega_2 HM\sqcup\dots\sqcup\omega_l HM$. Can I explicitly calculate these $\omega_i$?

Answer 1

I believe that your set $\exp(\mathfrak{h})$ will always act transitively on $S^n$ and will contain a stabilizer of at least one point of $S^n$, which should imply that $\exp(\mathfrak{h})$ alone generates $K$. Or am I missing something?

Answer 2

The pair (SO(n), SO(n-1))) is a symmetric pair, hence in this case M is a maximal subgroup of SO(n). Thus the connected component of your subgroup must the whole group.