We study the quantum geometry of doubly degenerate energy levels in a four-dimensional parameter space. We find that the scalar quantum metric g and the Berry curvature F obey (tr g)²/16≥√(det g)≥ |2Tr(F∧ F)-Tr F∧ Tr F|/24. The first inequality characterizes the anisotropy in the metric. The second determinant inequality measures the self-duality of the traceless SU(2) part of the curvature under Hodge star operation and the algebraic closedness of the inter-level polarization amplitudes under SU(2) rotations in the doubly degenerate levels. The saturation of the determinant bound imposes a quaternionic Cauchy-Riemann equation, analogous to the complex analyticity imposed by ideal-band conditions in two-dimensional Chern insulators. As examples, four-band Dirac Hamiltonians automatically saturate the determinant bound and possess a topological zero in tr(F∧ F). We compare the differences between Kramers degeneracy and ordinary U(2) degeneracy. In addition to the non-Abelian geometric bound, the latter also obeys an independent first-Chern bound.