I am trying to find the complexification of $\mathfrak{sl}(2,\mathbb{R})$. What I know about complexification is pretty much in Hall's book, Lie groups, Lie algebras and Representations (page 65).
Here is given the definition of the complexification $V_{\mathbb{C}}$ of a real vector space V, as the space of linear combinations $v_1 + iv_2$ with $v_1$, $v_2$ in V. Where we can consider this space over a real or complex field in the natural way: $$\lambda(v_1 + iv_2)=\lambda v_1 + i \lambda v_2 \hspace{1cm} \lambda \in \mathbb{R}$$ $$i(v_1 + iv_2)=-v_2+iv_1$$
Considered this, when I have to calculate the complexification of $\mathfrak{sl}(2,\mathbb{R})$, I'd say I should distinguish the case of real or complex field.
Given the base of the space: $$\mathfrak{sl}(2,\mathbb{R})=<T_1, T_2, T_3>$$ $\hspace{4cm}$ $T_1= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $ $\hspace{1cm}$ $T_2= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $ $\hspace{1cm}$ $T_3= \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} $
I'd say that the complexification over the real field is given by $$\mathfrak{sl}(2,\mathbb{R})_{\mathbb{C}}=<T_1, T_2, T_3,iT_1, iT_2, iT_3>$$ and so over the real field $\mathfrak{sl}(2,\mathbb{R})_{\mathbb{C}}=\mathfrak{sl},(2,\mathbb{C})$. But in the case of the complexification over $\mathbb{C}$ I can't find a solution as $T_j$ and $iT_j$ are not anymore linear indipendent, so the base it seems to be given just by ${T_1, T_2, T_3}$, hence $\mathfrak{sl}(2,\mathbb{R})_{\mathbb{C}}=\mathfrak{sl}(2,\mathbb{R})$ over $\mathbb{C}$, but I am quite sure this is wrong. In the end my problem is that I can't say what $\mathfrak{sl}(2,\mathbb{R})_{\mathbb{C}}$ over the complex field is. Thanks a lot for the help