Breaking wavefunction into pieces

[copernicus1]

Sometimes we see the wave function broken into specific parts, like $$\psi_{\rm total}=\psi_{\rm space}\psi_{\rm spin}\psi_{\rm isospin}.$$ What determines which parts you include in writing down the wavefunction? For instance, why or why wouldn't we include the isospin part? This has important consequences because each part has a symmetry and has implications when trying to construct wavefunctions for identical bosons or fermions.

 

[Chopin]

If you're dealing with an operator which acts symmetrically on part of the wavefunction, then there's no point in writing that part out...e.g. if your Hamiltonian isn't coupled to the spin of the particle, then you don't have to worry about its spin indices, because they'll go out the same they came in. Technically speaking, you can do this whenever the generators of the symmetry group commute with the operator in question.