Embedding of $CP^2/CP^1$ into euclidean space

It is a standard exercise in embedding theory to show that $ S^3 \to \mathbb{R}^4$ given by $ (x,y,z) \mapsto (x^2-y^2,xy,xz,yz)$ induces an embedding $ \mathbb{R}P^2 \to \mathbb{R}^4$ . Since $ \mathbb{R}P^2/\,\mathbb{R}P^1 \cong \mathbb{R}P^2$ , the previous map gives an embedding of $ \mathbb{R}P^2/\,\mathbb{R}P^1$ into $ \mathbb{R}^4$ .

Is there a nice embedding of $ \mathbb{C}P^2/\,\mathbb{C}P^1$ into $ \mathbb{R}^8$ ?