An exercise asked to compute the axis and the angle of the rotation $THTH$.
$(S.1)$ is easy to understand by using the identity $exp(i\theta A) = \cos(\theta) \mathbb1 + i \sin(\theta) A$ for $A$ s.t. $A^2 = \mathbb1$.
What I don't get is how one can deduce from the expression of $U_1$ its axis and angle (and even that it is a rotation? I guess it is deduced from the expression $THTH$).
The only way I see to find the axis is to compute the kernel of $(U_1-\mathbb1)$ in the basis $(|0>, |1>)$, then express it on the Bloch sphere. I have no idea how to find the angle.
Is there a result that uses the fact that $I, X, Y, Z$ is a base for the hermitian operators, and directly gives the axis and angle as a function of the coefficients of the matrix in this base?