Real congruences of complex subspaces of 2 × 2 symmetric complex matrices

Consider complex subspaces of symmetric 2 × 2 complex matrices under real congruences (sending M to AMAT with A real and invertible). The orbits for dimension one were classified (in different language) two decades ago. Those for dimension two were recently classified by Coffman. It turns out that these two problems are in fact dual to each other, and Coffman’s classification yields a simpler expression for the orbits in dimension one.