It is well known that the automorphism group of exceptional Jordan algebra $\mathcal{h}_{3}(\mathbb{O})$ is the exceptional Lie group $F_{4}$. I am trying to understand the automorphism group of the Jordan Algebra of Hermitian matrices $h_{3}(\mathbb{F})$, $\mathbb{F} = \mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$. My suspicion is that at least the connected components are $\mathrm{SO}(3)$, $\mathrm{SU}(3)$ and $\mathrm{Sp}(3)$, respectively. My calculations, using the matrix model of projective spaces, result in

$$\mathbb{R}\mathrm{P}^{2} \simeq G_{\mathbb{R}} / \mathrm{SO}(2)$$ $$\mathbb{C}\mathrm{P}^{2} \simeq G_{\mathbb{C}} / \mathrm{SU}(2)$$ $$\mathbb{H}\mathrm{P}^{2} \simeq G_{\mathbb{H}} / \mathrm{Sp}(2)$$

where $G_{\mathbb{F}}$ are the desired groups, based on theorem 14.99 of Spinors and Calibration (F. Harvey). Of course, the above candidates lies in the desired groups by an action of the form $g(A) = gA\overline{g}^{\mathrm{t}}$. I am trying to compare with: $$ \mathbb{R}\mathrm{P}^{2} \simeq \mathrm{O}(3) / \mathrm{O}(2) \times Z_{2}$$ $$\mathbb{C}\mathrm{P}^{2} \simeq \mathrm{U}(3) / \mathrm{U}(2) \times U(1)$$ $$\mathbb{H}\mathrm{P}^{2} \simeq \mathrm{Sp}(3) / \mathrm{Sp}(2) \times \mathrm{Sp}(1)$$ References are welcome.


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