I'm studying geometry and am having trouble with an exercise problem. As a disclaimer, the material is in Korean and there might be some inaccurate things I got wrong when I translated them over to English.
Anyway, I learned that a Hopf map (or Hopf fibration) $H: \Bbb{S}^3 \rightarrow \Bbb{S}^2$ can be defined as:
$$ \begin{align} H(x, y, z, t) & = (x^2 + y^2 - z^2 - t^2,\ 2(xt + yz),\ 2(yt - xz)) \\ H(z_1, z_2) & = (\vert z_1 \vert^2 - \vert z_2 \vert^2,\ 2z_1 \bar{z_2}) \end{align} $$
where the first equation is where we define $\Bbb{S}^n$ to be:
$$ \begin{align} \Bbb{S}^3 & = \left\{(x, y, z, t) \in \Bbb{R}^4 : x^2 + y^2 + z^2 + t^2 = 1 \right\} \\ \Bbb{S}^2 & = \left\{(x, y, z) \in \Bbb{R}^3 : x^2 + y^2 + z^2 = 1 \right\} \end{align} $$
and the second equation is:
$$ \begin{align} \Bbb{S}^3 & = \left\{(z_1, z_2) \in \Bbb{C}^2 : \vert z_1 \vert^2 + \vert z_2 \vert^2 = 1 \right\} \\ \Bbb{S}^2 & = \left\{(t, z) \in \Bbb{R \times C} : t^2 + \vert z \vert^2 = 1 \right\} \end{align} $$
The exercise problem starts by stating we can also define Hopf maps with quaternions by first defining $\Bbb{S}^3$ and $\Bbb{S}^2$ as follows:
$$ \begin{align} \Bbb{S}^3 & = \left\{q = x + y\mathbf{i} + z\mathbf{j} + k\mathbf{k} \in \Bbb{H} : \Vert q \Vert = 1 \right\} \\ \Bbb{S}^2 & = \left\{p = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \in \Bbb{H} : \Vert p \Vert = 1 \right\} \\ \\ H(q) & = M_q(\mathbf{i}) = q\mathbf{i} \bar{q} \end{align} $$
and what I have to do is prove that this "image" is a Hopf map.
My approach
First define:
$$ \begin{align} q & = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \\ \bar{q} & = a - b\mathbf{i} - c\mathbf{j} - d\mathbf{k} \end{align} $$
Calculating $q\mathbf{i}\bar{q}$ gives us:
$$ \begin{align} q\mathbf{i} & = -b + a\mathbf{i} + d\mathbf{j} - c\mathbf{k} \\ q\mathbf{i}\bar{q} & = (-b + a\mathbf{i} + d\mathbf{j} - c\mathbf{k})(a - b\mathbf{i} - c\mathbf{j} - d\mathbf{k}) \\ & = -ab + b^2\mathbf{i} + bc\mathbf{j} + bd\mathbf{k} \\ & \phantom{=} + a^2\mathbf{i} + ab - ac\mathbf{k} + ad\mathbf{j} \\ & \phantom{=} + ad\mathbf{j} + bd\mathbf{k} + cd - d^2 \mathbf{i} \\ & \phantom{=} -ac\mathbf{k} + bc\mathbf{j} - c^2\mathbf{i} - cd \\ & = (a^2 + b^2 - c^2 - d^2) \mathbf{i} + 2(ad + bc)\mathbf{j} + 2(bd - ac)\mathbf{k} \end{align} $$
This is where I got stuck. How am I supposed to "prove it's a Hopf map?"