I am interested in knowing various examples of non-split (added hypothesis reductive) **reductive** linear algebraic groups. In particular, I would like to collect the following examples in my counter-example toolbox.

Given an integer $n>2$, an extension $E/F$ of degree $n$ and a connected reductive $F$-group $G$ that splits over $E$ (added following Prof. Humphreys' comment:)

~~and $E$ is the smallest such extension~~minimally;Given a finite set $S$ of primes, a connected reductive group $G$ defined over $\mathbb Q$ which is quasi-split precisely over $\mathbb Q_v$ for $v \not\in S$ and not otherwise.

The motivation for this question was that I was reading Tits' article on Buildings in Corvallis and wanted to explicitly compute stuff about non-split groups. This prompted me to look for nonsplit groups, hence this question.

[I would like to make this question community-wiki since I am really asking 2 questions here and there is no unique answer, but for some reasons I can't find the CW box. Perhaps I need more reputation. ]