Generalizing Rules of Three
This behavior has been seen in some notable cases. Kirillov [3] shows that elementary symmetric polynomials in noncommuting variables commute (and, in some cases, all Schur functions) when elementary symmetric polynomials of degree at most three commute when restricted to at most three of the variables. Generalizing this, Blasiak and Fomin [1] give a wider theory for rules of three of generating functions over rings via rules of three in sums and products.