Consider the Hausdorff space $S^4 = \mathbb{HP}^1$ and the topological group $Sp(1)$ (identified with the unit quaternions), the cover $\{V_j\}_{j \in J} = \{U_1, U_2\}$ of $\mathbb{HP}^1$ where $$U_1= \{[q^1, q^2] \in \mathbb {HP}^1; q^1 \neq 0 \}\ \ \text{and} \ \ U_2= \{[q^1, q^2] \in \mathbb {HP}^1; q^2 \neq 0 \}$$
and the transition functions $g_{ji} : U_i \cap U_j \to Sp(1)$ for $i,j= 1,2$ are determined by $$g_{12} ([q^1,q^2]) = (q^2/|q^2|) (q^1/|q^1|)^{-1}$$
By the Reconstruction Theorem there exists an $Sp(1)$-bundle over $S^4 = \mathbb {HP}^1$ with transition functions $\{g_{11}, g_{12}, g_{21}, g_{22}\}$.
On the other hand consider the usual Hopf bundle $(S^7,\mathbb{HP}^1, \mathcal P, S^3 )$ where $\mathcal P: S^7 \to \mathbb{HP}^1$ is given by $\mathcal P (q) = [q] = [q^1,q^2]$ (restriction of the quotient map that determines the equivalence relation $\sim$ in $\mathbb{H}^2 -\{0\}$ to the sphere $S^7$) where the transition functions $g'_{ji} : U_i \cap U_j \to Sp(1)$ for $i,j= 1,2$ are determined by $$g'_{12} ([q^1,q^2]) = (q^1/|q^1|) (q^2/|q^2|)^{-1}$$
Question:
Show that the underlying locally trivial bundle for the $Sp(1)$-bundle over $S^4 = \mathbb {HP}^1$ is the same of the Hopf bundle, but the action of $Sp(1)$ on $S^7$ is different. Describe the action.
Thoughts:
$1)$ From the Reconstruction Theorem we have that the total space $P$ is the set of all equivalence classes $$[x,g,j] = \{(x,g_{kj}(x)g, k) ; k \in \{1,2\} \ \ \text{and} \ \ x \in U_j \cap U_k\}$$
I couldn't show that $P$ is the sphere $S^7$. The local trivializations for the $Sp(1)$-bundle are $$\begin{align}\Phi_j : U_j \times Sp(1) &\to \mathcal P^{-1}
(U_j)\\ (x,g) &\mapsto [x,g,j]\end{align}$$
while for the Hopf Bundle we have
$$\begin{align}\Psi_1 : U_1 \times Sp(1) &\to \mathcal P^{-1}(U_1)\\ ([q], y) =([q^1,q^2],y) &\mapsto (|q^1|y, q^2(q^1)^{-1}|q^1|y)\end{align}$$
similar to $\Psi_2$.
I can't see how $\Phi_1 = \Psi_1$.
2) The action $\sigma : P \times Sp(1) \to P$ is given by $\sigma (p,h) = p \cdot h = [x,g,j] \cdot h = [x, gh, j]$. Which I couldn't describe on $S^7$. So far, I understand that $\sigma$ acts on values of the group $Sp(1)$ alone where the action $\sigma'$ given by $\sigma((q^1,q^2), h) = (q^1,q^2)\cdot h=(q^1 h, q^2 h)$, for any $q^1, q^2 \in \mathcal P^{-1} (x)$, where $x \in S^4=\mathbb {HP}^1$ acts on elements not necessarily from the group $Sp(1) = S^3$. Is this conclusion anywhere near of what is happening here?