In this paper the Lorentz transformation, considered as the composition of a rotation and a Lorentz boost, is decomposed into a linear combination of two orthogonal transforms. In this way a two-term expression of the Lorentz transformation by means of quaternions is proposed. An analytical solution to the problem of finding eigenvectors is given. The conditions for the existence of eigenvectors are specified. The quartet of eigenvectors, which occurs when the rotational axis is orthogonal to the velocity direction, is obtained for two cases: for the generic case of the Lorentz transformation and for the composition of the Lorentz boosts. It is shown that a quartet of eigenvectors exists for the composition of any Lorentz boosts. For the composition of boosts it is established that the half-sum (arithmetic mean) of the square roots of mutually inverse eigenvalues is found by combining half-rapidities of the original boosts according to the same cosine rule, which is used to combine the source rapidities into the resulting rapidity.

中文翻译：

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四元离子洛仑兹变换的特征向量四重奏

在本文中，将洛伦兹变换（被视为旋转和洛伦兹升压的组合）分解为两个正交变换的线性组合。以此方式，提出了通过四元数的洛伦兹变换的二项表达式。给出了寻找特征向量问题的解析解。指定了特征向量存在的条件。在以下两种情况下获得了特征向量的四元组：旋转轴与速度方向正交时发生的两种情况：对于Lorentz变换的一般情况以及对于Lorentz Boost的合成。结果表明，对于任何洛伦兹增强的组成，都存在一个四重特征向量。