I have recently finished studying a course on Lie algebras which included Cartan's classification. The main process we took when studying a particular semi-simple Lie algebra was to first complexify it, form a Cartan-Weyl basis, find the roots and then build up a representation of highest weight. I have a few questions regarding this
How does studying the complexified Lie algebra $\mathfrak{g}_\mathbb{C}$ help us understand the original Lie algebra $\mathfrak{g}$? The method of highest weight only works for the complexified Lie algebra, so what could roots and weights tell me about the original Lie algebra?
Do $ \mathfrak{g} $ and $\mathfrak{g}_\mathbb{C} $ exponentiate to the same Lie group?