I'm wondering whether the adjoint and the standard representations of su(2) (the lie algebra of SU(2)) are equivalent. I found this result for so(3) by showing that given the usual basis of so(3), F1, F2, F3, the lie bracket relations for basis elements are:
[F1, F2]=F3, [F1,F3]=F2, [F3,F2]=F1
Thus the form of matrices is preserved by the adjoint map and hence is equivalent to the standard representation. Same argument seems to work for su(2), but this equality is nowhere, to my knowledge, mentioned. Hence I'm worried that something is wrong with my approach.
The two representations have different dimensions (standard is $2$, adjoint is $3$), so they can't be equivalent in the normal definition of equivalence between representations. If what you mean is that they have the same commutation relations, then that's what expected from all representations of the same algebra.
(PS. I wanted to add this as a comment, but wasn't allowed.)