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We consider cosmological dynamics in Einstein-Gauss-Bonnet gravity. Our study reveals that the transition from Gauss-Bonnet Kasner regime to anisotropic exponential expansion (with expanding three and contracting extra dimensions) is stable with respect to breaking the symmetry within both three- and extra-dimensional subspaces. However, the details of the dynamics in D=2 and D⩾3 are different. The result obtained allows us to construct a scenario in D⩾3, where isotropisation of outer and inner subspaces is reached dynamically from rather general anisotropic initial conditions