The above are part of the articles and some background of it. And there is one claim in the articles, saying that " in order to show that no trivial ring can be formed, it is sufficient to show that no non-trivial sequence of the matrices $R_1,R_2,R_3,R_4$ is a permutation matrices". But why it only consider the permutation matrix? It may be possible that it gives another form of matrix and give the permutation of the original position vector. As we know, either one of the position vector $OA,OB,OC, OD$ can be expressed in a linear combination of another 3 vector.It seems possible to have another form matrices.The authur seem to ignore that case with no reasons provided.
Also,there are some points in the comment which i don't qutie understand. Any explaination is appreciated.