I am calculating the generators of the algebra
$$\mathfrak{u}(2)=\{S \in Mat(n,\mathbb{C}) | S^{\dagger}=-S \}$$ $$ S=\begin{pmatrix} a_{11} + i\,b_{11} & a_{12} + i\,b_{12} \\ a_{21} + i\,b_{21} & a_{22} + i\,b_{22} \end{pmatrix} \quad (a_{ii},b_{ii} \in \mathbb{R})$$
$$ \begin{pmatrix} a_{11} - i\,b_{11} & a_{21} - i\,b_{21} \\ a_{12} - i\,b_{12} & a_{22} - i\,b_{22} \end{pmatrix} = \begin{pmatrix} -a_{11} - i\,b_{11} & -a_{12} - i\,b_{12} \\ -a_{21} - i\,b_{21} & -a_{22} - i\,b_{22} \end{pmatrix} \quad $$ $$S=A+iB$$ $$A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \,\,\,\,\,\, B=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} $$
Considering $\sigma_i$ the Pauli matrices, it is known that an algebra is:
$$ \left(\ T^{1}, \ T^{2}, \ T^{3}, \ T^{4} \ \right) = \left(\ \frac{1}{2} \sigma_{1}, \ \frac{1}{2} \sigma_{2}, \ \frac{1}{2} \sigma_{3}, \ \frac{1}{2} \mathbb{I} \ \right) $$ $$\sum_{i=1}^{4}T^{i}= \frac{1}{2}\begin{pmatrix} 2 & 1-i \\ 1+i & 0 \end{pmatrix}$$
Did I make a miscalculations? If my derivation is right what is the relationship between the two?
You simply transcribed the solution for the generic Lie algebra element wrong; solving the defining constraint yields, by inspection, $$S= a_{21}\begin{pmatrix} 0& -1 \\ 1 &0 \end{pmatrix} +i \begin{pmatrix} b_{11} & b_{12} \\ b_{12} & b_{22} \end{pmatrix} \\ = -ia_{21} ~ \sigma_{2} +i\left ( b_{12}~\sigma_{1} + \frac{b_{11}-b_{22} }{2} ~\sigma_{3} + \frac{b_{11}+b_{22}}{2} ~\mathbb{I} \right) . $$