An Einstein manifold in four dimensions has some configuration of $SU(2{)}_{+}$ Yang-Mills instantons and $SU(2{)}_{-}$ anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group Spin(4) is a direct product of two groups $SU(2{)}_{\pm }$ and the vector space of 2-forms decomposes into the space of self-dual and anti-self-dual 2-forms. It explains why the four-dimensional spacetime is special for the stability of Einstein manifolds. We now consider whether such a stability of four-dimensional Einstein manifolds can be lifted to a five-dimensional Einstein manifold. The higher-dimensional embedding of four-manifolds from the viewpoint of gauge theory is similar to the grand unification of the Standard Model, since the group $SO\left(4\right)\cong \mathrm{Spin}\left(4\right)/Z_2=SU(2{)}_{+}\otimes SU(2{)}_{-}/Z_2$ must be embedded into the simple group $SO\left(5\right)=Sp\left(2\right)/Z_2$. Our group-theoretic approach reveals the anatomy of Riemannian manifolds quite similar to the quark model of hadrons in which two independent Yang-Mills instantons represent a substructure of Einstein manifolds.