A closed connected subgroup
$\Gamma $
of the reduced stabilizer
$S{\mathbb{G}}_{0}$
of a locally compact connected translation plane
$\left(P,\mathcal{\mathcal{L}}\right)$ is called parabolic, if it
fixes precisely one line
$S\in {\mathcal{\mathcal{L}}}_{0}$
and if it contains at least one compression subgroup. We prove that
$\Gamma $ is a semidirect
product of a stabilizer
${\Gamma}_{W}$,
where
$W$ is
a line of
${\mathcal{\mathcal{L}}}_{0}\setminus \left\{S\right\}$
such that
${\Gamma}_{W}$
contains a compression subgroup, and the commutator subgroup
${R}^{\prime}$ of the
radical
$R$ of
$\Gamma $. The stabilizer
${\Gamma}_{W}$ is a direct product
${\Gamma}_{W}=K\times \Upsilon $ of a maximal compact
subgroup
$K\le \Gamma $ and a compression
subgroup
$\Upsilon $. Therefore, we
have a decomposition
$\Gamma =K\cdot \Upsilon \cdot N$
similar to the
Iwasawadecomposition of a reductive Lie group.
Such a “geometric
Iwasawadecomposition”
$\Gamma =K\cdot \Upsilon \cdot N$ is possible
whenever
$\Gamma \le S{\mathbb{G}}_{0}$
is a closed connected subgroup which contains at least one compression subgroup
$\Upsilon $. Then the set
$\mathcal{S}$ of all lines
through
$0$
which are fixed by some compression subgroup of
$\Gamma $ is homeomorphic to a
sphere of dimension
$dimN$.
Removing the
$\Gamma $invariant
lines from
$\mathcal{S}$ yields
an orbit of
$\Gamma $.
Furthermore, we consider closed connected subgroups
$N\le S{\mathbb{G}}_{0}$
whose Lie algebra consists of nilpotent endomorphisms of
$P$. Our main result states
that
$N$ is a direct product
$N={N}_{1}\times \Sigma $ of a central subgroup
$\Sigma $ consisting of all shears in
$N$ and a complementary normal
subgroup
${N}_{1}$ which contains
the commutator subgroup
${N}^{\prime}$
of
$N$.
