I have a simple question. I take the example of $su(2)$
We have :
$$ su(2)=\{ a \in \mathcal{M}_N(\mathbb{C}) / ia=\sum_{i=1}^3 \alpha_i~\sigma_i, \alpha_i \in \mathbb{R} \}$$
Where $\sigma_i$ is a Pauli matrix.
We often define :
$$ isu(2) = \{ T=\sum_{i=1}^3 \alpha_i~\sigma_i, \alpha_i \in \mathbb{R} \}$$
And we can take the basis : $ T_a = \frac{\sigma_a}{2} $.
When taking this basis we end up with :
$$ [T_a, T_b]=i f_{abc} T_c $$ where $f_{abc} \in \mathbb{R} $
The thing that I don't understand is that now we don't have the stability of $isu(2)$ under the commutator. Indeed we have $[T_a, T_b]$ that is a complex sum of matrices $T_k$ (it should be a sum with real coefficients to ensure the stability). Is it because $isu(2)$ is not a Lie algebra but just a space isomorphic to it ?
