| 31. | The structure constants C ^ { abc } quantify the lack of commutativity, and do not vanish.
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| 32. | First assume that ( 1 ) associates from either the left or the right, then prove commutativity.
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| 33. | This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent.
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| 34. | Commutativity applies; thus 3? must equal 7?, and we can count 7, 14, 21.
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| 35. | An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions.
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| 36. | A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that ( ) above is true.
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| 37. | Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism.
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| 38. | Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.
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| 39. | As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting.
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| 40. | Maybe one can get past that using the non-commutativity of the group operation, as opposed to ring addition?
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