  Assume that R is a semi-local regular ring containing an infinite perfect
field, or that R is a semi-local ring of several points on a smooth scheme over
an infinite field. Let K be the field of fractions of R. Let H be a strongly
inner adjoint simple algebraic group of type E_6 or E_7 over R, or any twisted
form of one of the split groups of classical type O^+_{n,R}, n>=4; PGO_{n,R},
n>=4; PSp_{2n,R}, n>=2; PGL_{n,R}, n>=2. We prove that the kernel of the map
H^1_{et}(R,H)-> H^1_{et}(K,H) induced by the inclusion of R into K is trivial.
This continues the recent series of papers by the authors and N. Vavilov on the
Grothendieck--Serre conjecture.
