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Does that mean that all the matrices of an integer-spin rep of SU(2) are identical to the matrices of the corresponding spin rep of SO(3)?

Say, the j=1 rep of SU(2) has three 3x3 matrices, so has the j=1 rep of SO(3). Are the matrices for SU(2) identical to those for SO(3)? But their are different groups, so that can't be right.

Or are the matrices identical only for the Lie algebras su(2), so(3)?

THANK YOU