Use that the adjoint representation for \(\mathrm{SU}(2)\) is three-dimensional and that generators of the adjoint representation are the structure constants of the group to construct a 3D matrix representation of SU(2). Verify explicitly that the resulting matrices satisfy the \(\mathrm{SU}(2)\) algebra. Using the standard methods of matrix algebra, transform this set of matrices to a new set \(T_{1}, T_{2}, T_{3}\), where \(T_{3}\) is diagonal.