# Conjugacy classes of SO3 and O3

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
The answer you accepted has been removed. –  joriki May 20 '13 at 21:48

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