Editing Lie–Kolchin theorem (section)

== Counter-example ==

If the field ''K'' is not algebraically closed, the theorem can fail. The standard [[unit circle]], viewed as the set of [[complex number]]s <math> \{ x+iy \in \mathbb{C} \mid x^2+y^2=1 \} </math> of absolute value one is a 1-dimensional [[abelian group|commutative]] (and therefore solvable) [[linear algebraic group]] over the [[real number]]s which has a 2-dimensional representation into the [[special orthogonal group]] SO(2) without an invariant (real) line.  Here the image <math>\rho(z)</math> of <math>z=x+iy</math> is the [[orthogonal matrix]]

: <math>\begin{pmatrix} x & y \\ -y & x \end{pmatrix}.</math>