The orthogonality principle for Osserman manifolds

We introduce a new potential characterization of Osserman algebraic curvature tensors. An algebraic curvature tensor is Jacobi-orthogonal if \(\mathcal{J}_XY\perp\mathcal{J}_YX\) ${\mathcal{J}}_{X}Y\perp {\mathcal{J}}_{Y}X$ holds for all \(X\perp Y\) $X\perp Y$ ,where \(\mathcal{J}\) $\mathcal{J}$ denotes the Jacobi operator.We prove that any Jacobi-orthogonal tensor is Osserman, while all known Osserman tensors are Jacobi-orthogonal.