Great Circles in $SU_{2}$

So I am working on the proof that all great circles in $SU_{2}$ (circles of radius 1) are a coset of a longitude, and I am unsure what a great circle looks like in matrix form.

Clearly any point on the 3-sphere takes the form $$\begin{bmatrix} x_{0}+x_{1}i & x_{2}+x_{3}i \\ -x_{2}+x_{1}i & x_{0}-x_{1}i \end{bmatrix},\ \ \ \ x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$$ and this specific point is essentially on infinitely many great circles, but how do I arbitrarily define (parameterize) matrix representing a great circle on the 3-sphere?

Artin's book does not mention much about arbitrary great circles on the matrix, and I really only understand how to paramterize a latitude or longitude in matrix form.

Consider an arbitrary $A\in SU_{2}$ on a great circle $G$. Then if we consider the set $A^{-1}G$, since $A\in G,$ then this set must contain the north pole ($I\in SU_{2}$). $G=SU_{2}\cap V$, where $V$ is a $2$-dimensional vector space. So $$A^{-1}G=A^{-1}SU_{2}\cap A^{-1}V=SU_{2}\cap A^{-1}V.$$ So if you can show that $A^{-1}V$ is a $2$-dimensional vector space, then $A^{-1}G$ will be a $2$-dimensional space in $SU_{2}$ which contains $I$, and so then $A^{-1}G$ will be a longitude. And then you can express $G$ as $$G=AA^{-1}G=A(A^{-1}G).$$