Let G be an algebraic group over a field k, and let ψ be an action of the multiplicative group km of k on G by automorphisms. We say ψ is an elementary action if it has only the weights 0, ±1; more precisely, if there exist subgroups H, U+, U- of G such that (i) H is fixed under ψ, (ii) U+ and U+ are vector groups and (iii) Ω = U+. H . U+ is open in G, and (iv) G is generated by H, U+, U+. This situation is characteristic for the complexifications of the automorphism groups of bounded symmetric domains (see, e.g., [9, 16]). A typical example is G = GLnwith (matrices being decomposed into 4 blocks) ψ given by