  Using the Lie derivative of the metric we define a class of Lie algebras of
vector fields by generalising the concept of Killing vectors. As a Lie algebra
they define locally a group action on the pseudo-Riemannian manifold through
exponentiation. The motivation behind studying these infinitesimal group
actions is the investigation of $\mathcal{I}$-degenerate pseudo-Riemannian
spaces, i.e., spaces having identical polynomial curvature invariants. In
particular, we show that all the known examples of $\mathcal{I}$-degenerate
pseudo-Riemannian spaces possess such vector fields.
