I need to obtain an explicit expression of the complexification of a real Lie algebra (in the sense of obtaining a form for all of its matrices). In the middle of my research I met the real Lie algebra:
$$\mathfrak{n} = \{\Bigg(\begin{array}{ccc} -2b & 5b & w \\ -5b & 3b & w \\ -\bar{w}& \bar{w} & -b \end{array} \Bigg), \text{ where } w \in \mathbb{C}, \ b \in \mathbb{R}\},$$ which obviously has real dimension 3. I need to find the complexification of this Lie algebra explicitly realized as 3x3 complex matrices. Is it possible? How is it done? I think I can`t simply write $$\mathfrak{n}_{\mathbb{C}} = \{\Bigg(\begin{array}{ccc} -2b & 5b & w + z \\ -5b & 3b & w + z \\ -\bar{w} - \bar{z}& \bar{w} + \bar{z} & -b \end{array} \Bigg), \text{ where }w,\ z,\ b \in \mathbb{C}\}.$$
My almost nil experience with complexification is that the complexification of $\mathfrak{sl}(n, \mathbb{R})$ is $\mathfrak{sl}(n, \mathbb{C})$, but this case seems more involved.
$$\mathfrak{n}_{\mathbb{C}}=\left\{\begin{pmatrix}-2b&5b&w\\-5b&3b&w\\-z&z&-b\end{pmatrix}\,\middle|\,b,z,w\in\mathbb{C}\right\}$$