  The two-jet of the curvature tensor at some point of a pseudo-Riemannian
manifold is called Einstein if the Ricci tensor is a multiple of the metric
tensor at the given point and additionally its first two covariant derivatives
vanish there. Following the Jet Isomorphism Theorem of pseudo-Riemannian
geometry, we derive necessary and sufficient conditions for the Einstein
property in terms of the symmetrization of the given two-jet (i.e. in terms of
the Jacobi operator and its first two covariant derivatives along arbitrary
geodesics emanating from the given point). A central role is played by the
Weitzenb\"ock formula for the Laplacian d delta + delta d acting on sections of
the vector bundle of algebraic curvature tensors. As an application, we study
linear Jacobi relations of order two on Einstein manifolds.
