I'm having some trouble understanding how rotations of $\mathbb{R}^3$ correspond to antipodal pairs of unit quaternions.
In section 1.5 of his Naive Lie Theory, John Stillwell proves the theorem that conjugation by $\cos \theta + u \sin \theta$, where $u$ is a purely imaginary unit quaternion, results in a rotation of the purely imaginary quaternions by $2 \theta$ about $u$.
He then says that
This theorem shows that every rotation of $\mathbb{R}^3$, given by an axis $u$ and angle of rotation $\alpha$, is the result of conjugation by the unit quaternion $$t = \cos \frac{\alpha}{2} + u \sin \frac{\alpha}{2}.$$ The same rotation is induced by $-t$, since $(-t)^{-1}s(-t) = t^{-1}st$. But $\pm t$ are the only unit quaternions that induce this rotation, because each unit quaternion is uniquely expressible in the form $t = \cos \frac{\alpha}{2} + u \sin \frac{\alpha}{2}$, and the rotation is uniquely determined by the two (axis, angle) pairs $(u, \alpha)$ and $(-u, -\alpha)$. The quaternions $t$ and $-t$ are said to be antipodal, because they represent diametrically opposite points on the 3-sphere of unit quaternions.
Thus the theorem says that rotations of $\mathbb{R}^3$ correspond to antipodal pairs of unit quaternions.
I don't understand what Stillwell means when he says that "each unit quaternion is uniquely expressible in the form $t = \cos \frac{\alpha}{2} + u \sin \frac{\alpha}{2}$"? Indeed, this seems to be false, since $$\cos \frac{\alpha}{2} + u \sin \frac{\alpha}{2} = \cos \frac{-\alpha}{2} + (-u) \sin \frac{-\alpha}{2}, \tag{*}$$ but $\alpha \neq -\alpha$ and $u \neq -u$.
I also don't understand what he means by "the rotation is uniquely determined by the two (axis, angle) pairs $(u, \alpha)$ and $(-u, -\alpha)$". Surely only one of these is sufficient to determine the rotation in question?
I also note that on the Wikipedia page for Quaternions and spatial rotations, it is stated that the fact each rotation is represented by two distinct unit quaternions
reflects the fact that each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction (a so-called double cover).
This confuses me since this "double cover" property of rotations seems to correspond more to $(*)$ than to the fact that conjugation by $-t$ gives the same rotation as conjugation by $t$. Indeed, writing
$$ \begin{align} -t &= -\cos \frac{\alpha}{2} - u \sin \frac{\alpha}{2} \\ &= \cos(\frac{\alpha}{2} + \pi) + u \sin(\frac{\alpha}{2} + \pi) \\ &= \cos \frac{\alpha + 2 \pi}{2} + u \sin \frac{\alpha + 2 \pi}{2} \end{align}$$
it appears $-t$ corresponds to the rotation $(u, \alpha + 2 \pi)$ rather than $(-u, -\alpha)$ as intimated by this remark. (I say this somewhat metaphorically -- I am aware that as objects $(u, \alpha + 2 \pi)$ and $(-u, -\alpha)$ are in fact identical.)
How does this correspondence actually work?