Let G be a reductive group over a commutative ring R. We say that G has isotropic rank ≥ n, if every normal semisimple reductive R-subgroup of G contains (Gm,R)n. We prove that if G has isotropic rank ≥ 1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A= R[x ₁ , … , x _n ] / I over R, the map HNis1(A,G)→HNis1(R,G) induced by evaluation at x ₁ = ⋯ = x _n = 0 , is a bijection. If k has characteristic 0, then, moreover, the map He´t1(A,G)→He´t1(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is ≥ 2 , and A is square-free, then K1G(A)=K1G(R), where K1G(R)=G(R)/E(R) is the corresponding non-stable K ₁ -functor, also called the Whitehead group of G. The corresponding statements for G= GL _n were previously proved by Ton Vorst.