 Adjoint Chevalley group

Fix a Chevalley basis and its integral lattice. The adjoint Chevalley group over a field is generated by the root operators obtained from the integral divided-power polynomials $x_{\alpha }(t)={\sum }_j{t}^j(\mathrm{ad}{e}_{\alpha }{)}^j/j!$ on that lattice. The coefficients are first computed over the integers and then reduced to the field; factorials need not be invertible there. This is the adjoint construction; other weight lattices give other isogeny forms of groups of Lie type.