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

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