arxiv.org/pdf/2201.04600.pdf claims to beat Mathematica on finding sequence formulas. Web app:

recur-env.eba-rm3fchmn.us-east-2.elasticbeanstalk.com

-Subscript[A, 111]^2 Subscript[B, 211]^2 +
  Subscript[A, 111]^2 Subscript[CC, 211]^2 -
  2 Subscript[A, 111] Subscript[A, 211] Subscript[CC, 111] Subscript[
   CC, 211] + Subscript[A, 211]^2 Subscript[CC, 111]^2 +
  Subscript[B, 211]^2 Subscript[CC, 111]^2;

Is it possible to represent this expression in vector-matrix form?

@dtn I mean, you can write it in terms of tensor operations likely. Here's part of the expression:

AA = {Subscript[A, 111], Subscript[A, 211]};
BB = {Subscript[B, 111], Subscript[B, 211]};
CC = {Subscript[C, 211], Subscript[C, 111]};

wAsymm = {{1, -1}, {-1, 1}};
Tr[
 (wAsymm*Outer[Times, AA, AA]) . Outer[Times, CC, CC]
 ]

That doesn't include any of the terms involving B. But really what matters is how you got to that expression since that probably has some kind of physical meaning in what you're modeling