Is there a quick way to see that $\mathbb{R}^8$ is the smallest space where $\mathbb{C}P^2$ can be embedded. I know that $\mathbb{C}P^2$ is an adjoint orbit of $SU(3)$, and therefore $SU(3)$ acts adjointly on an $\mathbb{R}^8$ vector, is this enough to prove the statement.
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