Completely Reducible SL(2)-Homomorphisms

Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms φ: SL₂ → G whose image is geometrically G-completely reducible-or G-cr-in the sense of Serre; the description resembles that of irreducible modules given by Steinberg's tensor product theorem. In case K is algebraically closed and G is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of φ to be geometrically G-cr; this plays an important role in our proof.