Reducible representations - choice of new basis

I was looking at this tutorial about reductions of representations. At first, the basis consists of four orbitals $$(s_N, s_1, s_2, s_3)$$ and the representations look like this

and we want to reduce the 3D $$\Gamma^{(3)}(g)$$ representation. Because its matrices aren't block-diagonal, we need to perform some similarity transform to get them into the convenient form. For that, we should choose a new appropriate basis, i.e. the linear combination of our orbitals.

Now the tutorial says that the appropriate new basis looks like this:

And yes, the basis is correct and the new block-diagonal matrices can be further derived using it.

But I have no idea, how did the author come with this specific basis.

So, how can I find the appropriated basis to reduce a reducible representation?

• I suspect from a purely mathematical point of view it’s highly non-trivial (mathoverflow.net/questions/199911). However, I suppose you can find the irreducible representations (using the reduction formula) then use the projection formula to get the form of the MOs. See e.g. slide 13 of goicoechea.chem.ox.ac.uk/resources/s2_presentation.pdf onwards. – orthocresol Oct 11 '18 at 23:04
• @orthocresol Thank you! Please, try to write down a more detailed answer, so I could accept it. – Eenoku Oct 11 '18 at 23:29