# Automorphism group of formally real Jordan algebras of hermitian matrices

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.