Invert single vector dimension using only addition and inversion

Question

I'm trying to perform an audio task, to try and isolate audio from mixed tracks. Specifically, I have two tracks with the same music but different sounds. Both are stereo. The only operations I can apply to them are inverting and mixing them. Doing so gets rid of the common parts and leaves on the difference.

There is a similar question: Isolating audio tracks through mixing that solves a similar-looking problem through a system of linear equations, however the conditions are different and I tried applying the answer given as well as the solution the OP applied but I can't get a meaningful answer out of it (Wolfram Alpha). I'm not sure if I've interpreted that solution correctly.

I've boiled the problem down to this:

• Track 1: i + j
• Track 2: i + k

i represents the common audio, j and k represent the audio unique to each track

I need to isolate either i, j or k. To do this, I believe it is sufficient to create an asymmetrically signed pair, or more explicitly one of these:

• -i + j
• i - j
• -i + k
• i - k

Since I can only invert a whole stereo track and mix tracks together, the only operations permitted are inverting the original vector pairs and adding them together (so it's allowed to have -i -j and -i -k but you can't directly go to -i +j etc.). Any new vectors created can also be inverted and added of course.

I'm thinking this is sort of a 3-dimensional geometric problem of adding and inverting vectors only, sort of an "only with a compass and straightedge" problem. In that representation, it would mean that under those constraints I have to produce a vector with exactly one dimension or one with exactly one negative and one positive dimension (such as -i + j). Either that or it's a system of linear equations like the other question but represented that way it seems there's no solution.

So far I've been unable to prove whether there is a solution (a series of steps to arrive from the original 2 to one of the needed 4) or find one by trial and error.

Is this possible given the constraints? Is there a way to prove whether this is possible at all?