I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow \mathbb{C}P^3$ is given explicitly by $(z,w)\rightarrow(z^3,z^2w,zw^2,w^3)$ in homogenous coordinates.

I was wondering if the same could be done with quaternionic projective spaces, i.e. is there a 'veronese type' embedding: $\mathbb{H}P^1\rightarrow\mathbb{H}P^n$ ??

(I know that the non-comutativity of quaternions make the veronese map ill defined)