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We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of $\\mathrm{\\mathop{SL}}(2,\\mathbb{C})^4$-orbits on $\\mathcal{H}_4$. 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We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of $\\mathrm{\\mathop{SL}}(2,\\mathbb{C})^4$-orbits on $\\mathcal{H}_4$. 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