The aim of this paper is to study the properties of the Möbius addition \(\oplus \) under the action of the gyration operator gyr[a, b], and the relation between \((\sigma ,t)\)-translation defined by the Möbius addition and the generalized Laplace–Beltrami operator \(\Delta _{\sigma ,t} \) in the octonionic space. Despite the challenges posed by the non-associativity and non-commutativity of octonions, Möbius addition still exhibits many significant properties in the octonionic space, such as the left cancellation law and the gyrocommutative law. We introduce a novel approach to computing the Jacobian determinant of Möbius addition. Then, we discover that the gyration operator is closely related to the Jacobian matrix of Möbius addition. Importantly, we determine that the distinction between \(a\oplus x\) and \(x\oplus a \) is a specific orthogonal matrix factor. Finally, we demonstrate that the \((\sigma ,t)\)-translation is a unitary operator in \(L^2 \left( {\mathbb {B}^8_t,d\tau _{\sigma ,t} } \right) \) and it commutes with the generalized Laplace–Beltrami operator \(\Delta _{\sigma ,t} \) in the octonionic space.

本文的目的是研究旋转算子gyr[a, b]作用下莫比乌斯加法\(\oplus \)的性质，以及\((\sigma ,t)\)-平移之间的关系由八元空间中的莫比乌斯加法和广义拉普拉斯-贝尔特拉米算子 \(\Delta _{\sigma ,t} \) 定义。尽管八元数的非结合性和非交换性带来了挑战，莫比乌斯加法仍然在八元空间中表现出许多重要的性质，例如左抵消律和陀螺交换律。我们引入了一种计算莫比乌斯加法的雅可比行列式的新方法。然后，我们发现回转算子与莫比乌斯加法的雅可比矩阵密切相关。重要的是，我们确定 \(a\oplus x\) 和 \(x\oplus a \) 之间的区别是特定的正交矩阵因子。最后，我们证明 \((\sigma ,t)\) 平移是 \(L^2 \left( {\mathbb {B}^8_t,d\tau _{\sigma ,t} } \right) \) 并且它与八元空间中的广义拉普拉斯-贝尔特拉米算子 \(\Delta _{\sigma ,t} \) 交换。