Möbius addition ⊕ and gyration operator gyr[a,b]play an important role in gyrogroup theory and hyperbolic geometry.In this paper,the Möbius addition and gyration operator are extended to the octonionic space.Although the multiplication of octonions is non-commutative and non-associative,the gyration operator can repair the missing commutativity of Möbius addition,resulting in the gyrocommutative law.In addition,the gyration operator gives rich content to Möbius addition on the octonion,for example,the left loop property and the left cancellation law.The Möbius coaddition derived from the Möbius addition and gyration operator can satisfy the general commutative law in the octonionic space,i.e.,a(田)b=b(田)a.
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