I am looking for a reference with definitions on what it means for an algebraic group to be split, quasisplit, and nonsplit. I would like to see some examples of the different "types".
Thanks, Tom
I am looking for a reference with definitions on what it means for an algebraic group to be split, quasisplit, and nonsplit. I would like to see some examples of the different "types". Thanks, Tom 


Besides the basic definitions and examples, you will find a concise description of the vocabulary needed to talk about linear algebraic groups over fields that are not algebraically closed in T. A. Springer's article titled Reductive Groups, which appears in Part I of Automorphic Forms, Representaions, and LFunctions (Proceedings of a conference in Corvallis), A.Borel and W.Casselman (editors), AMS, 1979. 


As far as references go, you can look in pretty much any book on algebraic groups, like Borel or Humphreys or PlatonovRapinchuk. Here is a standard example: The norm torus of the extension $\mathbb{C}/\mathbb{R}$ is the circle group, whose analytification is the compact Lie group $U(1)$. It is not split, since it is not isomorphic to the multiplicative group (whose analytification is $\mathbb{R}^\times$). It is quasisplit because all tori are quasisplit. One example of a nonquasisplit group is $U_{2,\mathbb{C}/\mathbb{R}}$. The compactness of the analytification obstructs the existence of a real Borel subgroup. 

