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Let $R = \mathbb Z[\sqrt{-1}]$ and $$\Omega = \{X \in GL_4(R) : X \overline X = I_4 \text{ or } -I_4 \},$$ where $\overline X$ is the complex conjugate matrix of $X$.

Two matrices $A, B \in \Omega $ are defined to be similar if $$B=U^{-1} A \overline U$$ for some $U \in GL_4(R)$.

Are the similarity classes of matrices in $\Omega$ classified?

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