Rakić duality principle

Pseudo-Riemannian manifold $(M,g)$ is Osserman if the characteristic polynomial of $\mathcal{J}$ is independent on unit pseudospheres, where $\mathcal{J}_X\colon Y\mapsto\mathcal{R}(Y,X)X$ is Jacobi operator. Rakić duality principle is a property $\mathcal{J}_X(Y)=\lambda Y\Rightarrow\mathcal{J}_Y(X)=\lambda X$. Osserman manifold and Rakić duality principle are in very close connection. Recent results show that in a Riemannian setting, manifold is Osserman if and only if Rakić duality principle holds.
We want to investigate this connection in a pseudo-Riemannian setting. Generalized duality principle becomes $\mathcal{J}_X(Y)=\varepsilon_X\lambda Y\Rightarrow\mathcal{J}_Y(X)= \varepsilon_Y\lambda X$, where $\varepsilon_X=g(X,X)$ is a norm of tangent vector $X$. We try to find an optimal domain for vectors $X$ and $Y$, starting with original where they are mutually orthogonal nonnull vectors.