Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)
Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.
