  By Perelman's $\mathcal L$-geodesic theory, we study the blow-down solutions
on a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^n, g)$
$(n\ge 4)$ with nonnegative curvature operator and positive Ricci curvature
away from a compact set of $M$. We prove that any $(n-1)$-dimensional compact
split ancient solution from the blow-down of $(M, g)$ is of type I. The result
is a generalization of our previous work from $n=4$ to any dimension.
