Can someone give me an example of an affine algebraic group over characterstic $0$, which consists of semisimple elements but is not commutative. Preferably given as a closed connected algebraic noncommutative subgroup of $GL_n(\mathbb{C})$ consisting soly out of diagonalizable matrices.
Edit: It turns out that the if we assume the group to be connected (arbitrary characteristic) then it will be commutative: The Borel subgroup will equal the group by Humpreys Linear Algebraic Groups 21.4 Proposition B. So, the group is solvable. But a solvable connected affine algebraic group is a semi-direct product of its unipotent part and a commutative connected quotient. See Humpreys 19.1. As the unipotent part (Jordan-Chevalley-decompostion) is assumed to be trivial, the group is commutative. As any matrix of finite order is diagonalizable over an algebraically closed field of characteristic zero, all nonabelian finite subgroups of $GL_n$ give an example to the question.