The aim of this paper is to derive consequences of a result of Götze and Zaitsev (2008). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (1985). We establish bounds for the rate of strong Gaussian approximation of sums of i. i. d. R^d-valued random vectors ξ_j having finite moments E {double pipe}ξ_j{double pipe}^γ, γ > 2. Bibliography: 13 titles.