This is a semisurvey paper, where we start by advertising
Tits’ synthetic construction (1953) of the hyperbolic plane
over the Cayley
numbers
and of its automorphism group which is the exceptional simple Lie group
. Let
be
the Iwasawa decomposition. Our contributions are:
Writing down explicitly the action of
on
in Tits’ model, facing the lack of associativity of
.
If
denotes the minimal parabolic subgroup of
,
characterizing
geometrically.
