A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.

Let $M = G/K$ be a rank-one symmetric space of the noncompact type. There are only three families of rank $1$ symmetric spaces:

1. hyperbolic $n$-space, corresponding to the Lie group $\operatorname{SO}(n,1)$.

2. complex hyperbolic $n$-space, corresponding to the Lie group $\operatorname{SU}(n,1)$.

3. quaternionic hyperbolic $n$-space, corresponding to the Lie group $\operatorname{Sp}(n,1)$.

There is also one exceptional example:

4. the Cayley upper half plane, corresponding to the Lie group $F_4^{-20}$.

Now, let $G = NAK$ be the Iwasawa decomposition of $G$. Does anyone know what is the Iwasawa decomposition ($N=?$ and $K=?$) of the four cases 1), 2) 3) and 4)?

A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.

Let $M = G/K$ be a rank-one symmetric space of the noncompact type. There are only three families of rank $1$ symmetric spaces:

1. hyperbolic $n$-space, corresponding to the Lie group $\operatorname{SO}(n,1)$.

2. complex hyperbolic $n$-space, corresponding to the Lie group $\operatorname{SU}(n,1)$.

3. quaternionic hyperbolic $n$-space, corresponding to the Lie group $\operatorname{Sp}(n,1)$.

There is also one exceptional example:

4. the Cayley upper half plane, corresponding to the Lie group $F_4^{-20}$.

Now, let $G = NAK$ be the Iwasawa decomposition of $G$. Does anyone know what is the Iwasawa decomposition ($N={?}$ and $K={?}$) of the four cases 1), 2), 3) and 4)?

A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie Group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.

Let $M = G/K$ be a rank-one symmetric space of the noncompact type. There are only three families of rank $1$ symmetric spaces:

1. hyperbolic $n$-space, corresponding to the Lie group $SO(n,1)$.

2. complex hyperbolic $n$-space, corresponding to the Lie group $SU(n,1)$.

3. quaternionic hyperbolic $n$-space, corresponding to the Lie group $Sp(n,1)$.

There is also one exceptional example:

4. the Cayley upper half plane, corresponding to the Lie group $F_4^{-20}$.

Now, let $G = NAK$ be the Iwasawa decomposition of $G$. Does anyone know how the Iwasawa decomposition (N=? and K=?) of the four cases 1), 2) 3) and 4)? thank you in advance

A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.

Let $M = G/K$ be a rank-one symmetric space of the noncompact type. There are only three families of rank $1$ symmetric spaces:

1. hyperbolic $n$-space, corresponding to the Lie group $\operatorname{SO}(n,1)$.

2. complex hyperbolic $n$-space, corresponding to the Lie group $\operatorname{SU}(n,1)$.

3. quaternionic hyperbolic $n$-space, corresponding to the Lie group $\operatorname{Sp}(n,1)$.

There is also one exceptional example:

4. the Cayley upper half plane, corresponding to the Lie group $F_4^{-20}$.

Now, let $G = NAK$ be the Iwasawa decomposition of $G$. Does anyone know what is the Iwasawa decomposition ($N=?$ and $K=?$) of the four cases 1), 2) 3) and 4)?