"real part" of complex differential operators on $Sl_{2}(\mathbb{C})$

Question

Let $D$ be a complex differential operator on the complex Lie-Group $SL_{2}(\mathbb{C})$. How is the "real part" of $D$ with regards to $SL_{2}(\mathbb{R})$ defined? Im currently reading a lecture series by Atiyah about characters of semisimple Lie-Groups and he considers $SL_{2}(\mathbb{R})$ as a real form of $SL_{2}(\mathbb{C})$ to prove certain equalities of differential operators on $SL_{2}(\mathbb{R})$ by "complexifying" them.

Answer 1

The complex Lie Group $\text{SL}_{2}(\mathbb{C})$ can be considered as a six dimensional Lie Group $\text{SL}_{2}(\mathbb{C})_{\mathbb{R}}$ The real and imaginary part are considered as independent real parameters.

The Lie Algebra of $sl_{2}(\mathbb{C})_{\mathbb{R}}$ turns out to be isomorphic to the Lorentz group.