In this paper, we show that $\mathrm{P}\mathrm{O}(2,1)$ can be represented by invertible split quaternions. We introduce the concepts of t-similarity and t-pseudosimilarity of split quaternions. By such two similarities, we obtain several equivalence relations in split quaternions. We also show that a and b are semisimilar if and only if $a^2$ and $b^2$ are similar.

中文翻译：

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分裂四元数中与O（2,1）有关的一些相似性类

在本文中，我们表明 $\mathrm{P}\mathrm{Ø}（2，\mathrm{1个}）$可以用可逆分裂四元数表示。我们引进的概念牛逼-similarity和牛逼分四元数-pseudosimilarity。通过这两个相似性，我们获得了分裂四元数中的几个等价关系。我们还表明，当且仅当且仅当a和b是半相似的${\mathrm{一种}}^2$ 和 $b^2$ 相似。