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Starting from the general expression:

i A(a x + b y + c z + ...) + j B(a' x + b' y + c' z + ...) + k C(a'' x +b'' y + c'' z + ...)

I'd like to rearrange it to obtain:

x(a i A + a' j B + a'' k C) + y(b i A + b' j B + b'' k C) + z(c i A + c' j B +c'' k C) + ...

Without using $x$, $y$, $z$ etc. as arguments for Collect, since there are too many parameters for that to be practical and each of $x$, $y$, $z$ etc. have a form that can't be known beforehand.

To clarify, the initial expression would be something like this: $$a_T \sum _{i=1}^6 \sum _{j=1}^6 \left(\text{phi}_{2,1} \text{Bin}[[i,j,5]] \text{Bout}[[i,j,4]]\right)+a_v \sum _{i=1}^6 \sum _{j=1}^6 \left(\text{phi}_{1,4} \text{Bin}[[i,j,6]] (-\text{Bout}[[i,j,1]])\right)+b_T \sum _{i=1}^6 \sum _{j=1}^6 \left(\text{phi}_{1,j} (-\text{Bout}[[2,3,i]]) \text{Bin}[[i,j,6]]\right)+b_V \sum _{i=1}^6 \sum _{j=1}^6 \left(\text{phi}_{2,j} \text{Bout}[[i,2,3]] \text{Bin}[[i,j,5]]\right)+c_V \sum _{i=1}^6 \sum _{j=1}^6 \sum _{k=1}^6 \left(\text{phi}_{5,k} \text{Bout}[[i,j,3]] \text{Bin}[[i,j,k]]-\text{phi}_{6,k} \text{Bout}[[i,j,4]] \text{Bin}[[i,j,k]]\right)$$

Every instance of $phi_{i,j}$, $bin_{i,j,k}$ and $bout_{i,j,k}$ (the elements of the Bin and Bout arrays) is a sum of symbolic terms so that in the end $x$, $y$ or $z$ may end up being something like $\Lambda \Omega \Sigma$. These final combinations are numerous enough that using them as an argument for a command is impractical (as writting them down would defeat the purpose of using the expression in the first place).

An example result of the above expression would be $$ 2 c_V \left(-8 \sqrt{2} \Xi ^0 \overline{\Xi ^0} K^{*0}\text{(0)}-12 \sqrt{2} \overline{\Xi ^0} K^0 \Xi ^0+5 \sqrt{6} \overline{\Sigma ^0} \omega _8\text{(0)} \Xi ^0\right) + b_T \left(8 \sqrt{2} \overline{\Xi ^0} K^0 \Xi ^0 + 3 \sqrt{6} \overline{\Sigma ^0} \omega _8 \text{(0)} \Xi ^0 - \sqrt{2} \overline{\Sigma ^0} \Xi ^0 \pi ^0 \right) + b_V\left(6 \sqrt{2} \unicode{f3b5} \unicode{f3b5} \Xi ^0 \overline{\Xi ^0} K^{*0}\text{(0)}+10 \sqrt{2} \overline{\Xi ^0} K^0 \Xi ^0-3 \sqrt{6} \overline{\Sigma ^0} \omega _8\text{(0)} \Xi ^0\right)$$

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Hello ! It is recommended that you also share the code that troubles you, not just expect people to solve the problems you present to them. –  Sektor Jan 29 at 16:16
    
I see that you extended your question. If my answer doesn't work for you, please give me additional input-output examples in Mathematica code. Thanks. –  Mr.Wizard Jan 30 at 0:32

2 Answers 2

Based on your sole example and no additional description I would use:

expr = i A (a x + b y + c z) + j B (a' x + b' y + c' z) + k C (a'' x + b'' y + c'' z);

co = Intersection @@ Variables /@ Cases[expr, _Plus, {2}]

Collect[expr, co]
{x, y, z}

x (a A i+B j a'+C k a'')+y (A b i+B j b'+C k b'')+z (A c i+B j c'+C k c'')

Note: C is a system Symbol.

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I'll bet there's a more elegant way to do this, but the following brute-force expression seems to do the trick as well

expr = i A (a x + b y + c z) + j B (a' x + b' y + c' z) +  k CC (a'' x + b'' y + c'' z)

Expand@expr //.
  {
   i___ j_ x + k_ x :> (i j + k) x,
   i___ j_ y + k_ y :> (i j + k) y,
   i___ j_ z + k_ z :> (i j + k) z
  }

which produces:

enter image description here

I have used CC instead of C for the reason Mr. Wizard pointed out.

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