The Iwasawa decomposition and the Bruhat decomposition of the automorphism group on certain exceptional Jordan algebra

Let ${\mathcal{J}}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is ${\mathrm{F}}_{4(-20)}$. By using some orbit types of ${\mathrm{F}}_{4(-20)}$ on ${\mathcal{J}}^1$, for ${\mathrm{F}}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima–Sekiguchi’s $K_{ϵ}-$Iwasawa decomposition, the Matsuki decomposition, and the Bruhat and Gauss decompositions.