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 algebra associated to $\mathfrak h$. What can we say about $HM$? When is it true that $HM=K$?

I know $K/HM$ is discrete and finite. This follows from the fact that $HM$ is open in $K$ and $K$ is compact. Thus there exists $e=\omega_1,\omega_2,\dots,\omega_l$ s.t. $K=HM\sqcup\omega_2 HM\sqcup\dots\sqcup\omega_l HM$. How do I explicitly calculate these $\omega_i$?

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 $\omega_i$?