Given the transformation of a $SU(2)$ triplet $\vec\phi$ $$\phi\to \exp{(-i\vec{T}\cdot\vec{\theta})}~\vec\phi\tag{1}$$ (in the question here by @physicslover) how does obtain the transformation of $\Phi\equiv\vec{\phi}\cdot\vec{\tau}$ given by $$\Phi\to e^{i\vec{\tau}\cdot\vec{\theta}/2}\Phi e^{-i\vec{\tau}\cdot\vec{\theta}/2}?\tag{2}$$ For a reference, look at the second equation of the answer by Cosmas Zachos.
The transformation by @physicslover was in terms of the $3\times 3$ matrix representations of the $\rm {SU(2)}$ generators $\vec{T}$. Then oracularly it became a transformation in terms of $\vec{\tau}$. Is there a systematic way to derive (2) from (1)?
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