Let R be a unital commutative ring. The elementary subgroup E_n(R) of the general linear group GL_n(R) is the subgroup generated by all unipotent elementary matrices in GL_n(R). For any isotropic reductive group scheme G over R, one can define analogs of unipotent elementary matrices, and the respective elementary subgroup E(R) of the group of R-points G(R). When G is a Chevalley group or R is a field, these are the so-called elementary root unipotents in G, parametrized by the roots in the root system of G, that were constructed in the 1950s and 1960s in the work of C. Chevalley, A. Borel, J. Tits, M. Demazure and A. Grothendieck. We will discuss the general case along the lines of the joint papers of the speaker and V. Petrov, and provide some applications to the study of the non-stable K_1-functor, or the Whitehead group, associated to G.