Diameter of $S{U}_2$ for a left-invariant axisymmetric Riemannian metric

We consider the Lie group $S{U}_2$ endowed with a left-invariant axisymmetric Riemannian metric. This means that a metric has eigenvalues $I_1 = I_2, I_3 > 0$. We give an explicit formula for the diameter of such metric. Other words, we compute the diameter of Berger’s sphere.