By Perelman's $L$-geodesic theory, we study blow-down solutions on a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^n, g)$ $(n\ge 4)$ with nonnegative curvature operator $\rm{Rm}$ and positive Ricci curvature away from a compact set of $M$. We prove a nonexistence result of compact split solutions of type II from the blow-down of $(M, g)$. As an application, we prove that $(M,g)$ with $\rm{Rm}\geq 0$ must be isometric to the Bryant Ricci soliton up to scaling, if the tangent flow of $(M,g)$ at $-\infty$ has a compact split ancient flow of codimentional one.