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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 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. – hardmath Oct 29 '12 at 13:30

Any two rotations through the same angle are conjugate: Rotate the first axis into the second, turn about the second axis, rotate back – the two rotations on the outside are inverses of each other, and the result is the same as if you'd turned about the first axis.

Any two rotations through different angles are not conjugate because their matrices have different traces $1+2\cos\phi$.

So two rotations are conjugate iff they're through the same angle.

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