So I have seen that every semi-simple complex lie algebra has a split and compact real form, where the compact real forms correspnding to semi-simple compact real lie algebras hence we can classify all possible complex semi simple algebras (and label them by their coreesponding compact semi simple real form).
Now Im wondering if it's possible to classify all real semi-simple lie algebras by real forms of the complex semi simple algebras. That is for any real semi simple $\mathfrak{g}$, does it correspond to a real form of one of the classified complex semi simple algebras $\mathfrak{g}^{\mathbb{C}}$?
Best Answer
Yes. See Theorem 6.94 of Knapp, Lie groups, beyond an introduction.
This states: if $\mathfrak{g}$ is a real simple Lie algebra then either its complexification $\mathbf{C} \otimes_{\mathbf{R}} \mathfrak{g}$ is simple or its complexification is not simple, $\mathfrak{g}$ is a complex simple Lie algebra, and its complexification is isomorphic to $\mathfrak{g} \oplus \mathfrak{g}$. In either case, $\mathfrak{g}$ is a real form of a semi-simple Lie algebra.