I'm currently studying the connection between $SU(2)$ and the $3$-sphere. A longitude on the sphere is of the form $QTQ^*$ where $Q \in SU(2)$ and $T = \{D_\lambda : \lambda \bar{\lambda} = 1 \} $, where $T$ is the group of diagonal matrices with entries: a complex number and its complex conjugate, such that determinant is $1$.
My understanding is that the subgroup $T$ of $SU(2)$ allows us to see explicitly the possible eigenvalues of our sets of transformations $QTQ^*$.
A proposition in my lecture notes says that $SO(2)$, a subgroup of $SU(2)$, can be identified with a longitude on the $3$-sphere.
In trying to make this more concrete I have written $QTQ^*$ and studied the entries more closely.
I find that for $QTQ^*$ to be a matrix of $SO(2)$ having entries contained in the reals, I need the following equations to hold:
$x_1^2+x_2^2-x_3^2-x_4^2= 0$
$x_1x_3 + x_2x_4 = 0$
and for the identification of $Q$ with some point on $S_3$:
$x_1^2+x_2^2+x_3^2+x_4^2= 1$
The problem is, these three equations seem to satisfy multiple solutions.
Then I can construct many different $Q$ to yield $QTQ^* \in SO(2)$ and the statement that any one $Q$ represents a longitude and $SO(2)$ is one such longitude does not make sense to me. I would have liked to find that only one such $Q$ is possible.
Is this the right way to think about this question? Are there more fruitful ways to perform the identification? If so, what is the central flaw of my reasoning? Thanks in advance.
Any element $q \in SU(2)$ defines a longitude $q^*Tq$ in $S^3$, but for any $t \in T \cong SO(2) \subset SU(2)$, we have $$(tq)^* T (tq) = q^* (t^* T t) q = q^* T q ,$$ so $t q \in SU(2)$ defines the same great circle that $q$ does, so there is at least a $1$-parameter family's worth of $q \in SU(2)$ that determine the same great circle.
Indeed, for all $\theta$, $$\pmatrix{x_1&x_2\\x_3&x_4} = \frac{1}{\sqrt{2}} \pmatrix{\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta}$$ is a solution to the quadratic system in $x_1, x_2, x_3, x_4$ given in the question statement.
In fact, direct computation shows that $SU(2)$ acts transitively on the space of $\mathcal L$ longitudes, and that the stabilizer of the longitude $T$ is $T \cong SO(2)$ itself. So we may identify $\mathcal L$ with the (right) homogeneous space $SO(2) \backslash SU(2)$, i.e., the longitudes with the right cosets $T \cdot q = q^* T q$, which in particular are conjugate copies of $SO(2)$.