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This is a semi-survey paper, where we start by advertising Tits' syntheticconstruction from \cite{Tits}, of the hyperbolic plane $H^2(Cay)$ over theCayley numbers $Cay$, and of its automorphism group which is the exceptionalsimple Lie group $G=F_{4(-20)}$. Let $G=KAN$ be the Iwasawa decomposition. Ourcontributions are: a) Writing down explicitly the action of $N$ on $H^2(Cay)$in Tits'model, facing the lack of associativity of $Cay$. b) If $MAN$ denotesthe minimal parabolic subgroup of $G$, characterizing $M$ geometrically.

The exceptional simple Lie group $F_{4(-20)}$, after J. Tits