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.

Any suggestions on how to think about this problem are appreciated!