I'm pretty confused by the following statement on page 6 in the book The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, which is meant to expain the fact that unit sphere $S^3$ in the quarternion algebra $\mathbb{H}$ is naturally associated with the group of rotation in $\mathbb{R}^3$:
I don't think the statemet is true at the very first place, $\alpha$ does not act trivially on $\mathbb{C}\alpha$, since by that we require for any complex number $z$, the equation $\alpha (z \alpha) \alpha^{-1}=z \alpha$, i.e. $$\alpha z \alpha^{-1}=z$$i.e. we need $\alpha$ commutes with any $z$. But this is not even true for $j \in S^3$ and $i \in \mathbb{C}$.
Can anyone tells me where I went wrong?
I don't think it's referring to a $1$ dimensional $\mathbb C$ subspace, I think it's referring to a 2 dimensional $\mathbb R$ subspace copy of $\mathbb C$.
Apparently "the complex plane spanned by $\alpha$" means $P_1=\mathrm{span}_\mathbb{R}(\{1, \alpha\})$ and the other plane refers to the plane perpendicular to $\alpha$ in the pure quaternions.
Indeed, conjugation on $P_1$ by $\alpha$ leaves the basis fixed, so it leaves all elements fixed.