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I'm trying to find the conjugacy classes of SO3 and O3. How do I do this?

SO3 consists of all rotations around any axis in three dimensions but how do I determine which are conjugate?

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    Construction of the similarity matrix for rotations with equal angles is described [in this Answer](http://math.stackexchange.com/a/207672/3111) to an earlier Question, solving $AX = XB$ for $X$ where all matrices are 3D rotations. Note that in 3D, for orthogonal $X$ either $X$ or $-X$ is a rotation. So the cases (special and general orthogonal groups) may be distinguished by checking the determinant and converted between by scalar multiplication by -1.2012-10-29

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