## 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 $$\mathbb{R}^8$$?