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So there are three permutations/rotations of these variables. And the three ternary as opposed to binary outputs -1,0,1. Very exciting. Therefore we have  \begin{equation}  [b,g,r^+]+[r^+,b,g]=[g,r^+,b] [b,g,r^+]+[r^+,b,g]=[g,r^+,b]=0  \end{equation}  If one switches two of the values side by side then it will lead to the complicated cube root coeficcients, it is not obvious if they cancel out under the same rules.We seek some mathematical differences and similarities between this contstruction and the regular binary commutator, for example, for some constant g \begin{equation}