Formula for the topological invariant for each of the symmetry classes

Is there a reference that systematically derives the topological invariant/winding number for all the ten symmetry classes in Altland and Zirnbauer's periodic table? For example, in this answer, there is the mention of the standard Chern number for the D class, but a slightly different winding number for the AIII class.

In general, given a gapped Bloch Hamiltonian that respects/breaks certain symmetries, how does one go about finding this winding number? Any insight would be helpful, even in specific cases. Thanks!