31. The structure constants C ^ { abc } quantify the lack of commutativity , and do not vanish. 32. First assume that ( 1 ) associates from either the left or the right, then prove commutativity . 33. This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent. 34. Commutativity applies; thus 3? must equal 7?, and we can count 7, 14, 21.35. An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions. 36. A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that ( ) above is true. 37. Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. 38. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears. 39. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. 40. Maybe one can get past that using the non-commutativity of the group operation, as opposed to ring addition?
