Why restrict to complex Lie algebras? I am taking a class about Lie algebras, where we introduced in the beginning the notion of a Lie algebra, but over time we restricted ourselves only to complex Lie algebras. Can someone of you tell me why? Are these complex Lie algebra so important because there is maybe a connection to complex manifolds? If yes, what about real Lie algebras, they could be connected to ''real'' manifolds. Or is the reason, that the underlying field is algebraically closed (but then: why not an arbitrary algebraically closed field)?
Best regards
 A: Beginner courses and textbooks often restrict to Lie algebras over $\mathbb C$ (or at least algebraically closed fields of characteristic $0$) because they are the basic case: They are the easiest to handle. This is largely due to linear algebra (and quadratic forms, which turn out to be intimately related to Lie algebras) i.e. matrix calculations being easiest over algebraically closed fields.
As a striking example, just for semisimple Lie algebras, the classification over $\mathbb C$ is done via root systems (and these, in turn, via Dynkin diagrams). That's a great classical result that is within reach for a one semester course right after linear algebra, and/or an introductory textbook on the topic.
(If the aim is strictly Lie theory i.e. Lie groups, usually from there it goes to compact Lie groups over $\mathbb R$ which happen to have a nice one-to-one relation to those complex Lie algebras.)
But if one is interested in Lie algebras, and not just the compact case of Lie groups, to get a classification of all semisimple Lie algebras over $\mathbb R$, or other characteristic $0$ fields, one has to use that complex classification / root system theory and kind of "enrich" it (e.g. search for "Satake diagrams" or "Satake-Tits diagrams" or "Tits index" which are enrichments of the above mentioned Dynkin diagrams). For the case of $\mathbb R$, look for example at Good source of references for the classification of real semisimple Lie algebras. For the case of other characteristic $0$ fields, I wrote my thesis about that, focussing more on $p$-adic fields (which to be honest was mostly collecting work scattered in the literature, I claim almost no originality; link to be found in the above link too). In there, I basically started by saying "we take the classification over algebraically closed fields for granted ..." and took it from there.
And even that does not even touch the case of positive characteristic. And even that is just semisimple Lie algebras. It goes wild from there.
Upshot: As often in higher math (and conversely to the impression one maybe gets in high school), over complex numbers a lot has been researched and many things are nicely classified. Over $\mathbb R$, we also know a lot, often relying on the complex case though or switching back and forth between real and complex scalars. But over other fields, like the rational numbers, problems get really tough.
A: Using complex algebra is easier because $\mathbb{C}$ is an algebraically closed field. Moreover, if you take a real Lie algebra $\mathfrak{A}$, you can associated a complex Lie algebra, which is the tensor product $\mathfrak{A}\otimes \mathbb{C}$, and you can study many of its properties like that.
Another thing is you are studying differential geometry, which is a geometry based on the topological properties of $\mathbb{R}^n$ (and $\mathbb{C}^n$). Using an arbitrary field would lose a lot of sense in this context.
A: Why not over an arbitrary algebraically closed field $K$?
The classification of simple (modular) Lie algebras over an arbitrary algebraically closed field of characteristic $p>0$ is still unknown for $p=2$ and $p=3$ and much more complicated for $p\ge 5$ than over the complex numbers.
In fact, a completely new type of simple algebras arises, of so called Cartan type. This is another reason, why a beginner class would start with a classification of complex Lie algebras.
For a reference, see for example this paper by Strade and Wilson.
