I have a function, br, that takes multiple integer arguments (which for syntactic ease I collect in a list) and is

• Antisymmetric in any two arguments:

  br[l_List] :> Signature[l] br[Sort[l]] 

• Obeys an algebraic relation:

   br[{i_,j_,m___}]br[{k_,l_,m___}] -  br[{i_,k_,m___}]br[{j_,l_,m___}] -  br[{i_,l_,m___}]br[{j_,k_,m___}] :> 0 

Perhaps abusively, I call this a “quadratic” constraint since each term is two “powers” of br (with different arguments).

Is there a good way to implement this second condition?

A sample expression that should reduce after two iterations of the second rule:

samp = (1/(br[{3, 6, 7}] br[{4, 6, 7}]))* ( br[{2, 6, 7}] br[{3, 6, 7}] br[{4, 5, 6}] +  br[{2, 6, 7}] br[{3, 4, 6}] br[{5, 6, 7}] +  br[{2, 3, 6}] br[{4, 6, 7}] br[{5, 6, 7}] ); 

The first condition is straightforward, but I have to by-hand define the second:

cleanBR = { br[l_List] :> Signature[l] br[Sort[l]] , br[{3, 6, 7}] br[{4, 5, 6}] :>  br[{3, 6, 4}] br[{7, 5, 6}] + br[{3, 6, 5}] br[{7, 4, 6}] , br[{2, 6, 7}] br[{3, 5, 6}] :>  br[{2, 6, 3}] br[{5, 7, 6}] + br[{2, 6, 5}] br[{7, 3, 6}] }; 

Then:

FixedPoint[(# //. cleanBR // Simplify) &, samp] (* br[{2,5,6}] *) 

Is there a clever way to do the br[___]br[___]:>___ that I currently resort to? I’m open minded about this and have aside from pattern matching considered algebraic elimination and some kind of Simplify[samp,extraEquations_List] but nothing is very robust.