If you remove this article OK, but still I doubt that you catch the most general aspect of curvature in the sense of Nomizu, Kulkarni and other modern writers. Look at the reference  where you find more references. In the references there (look also that one in Lecture Notes in Mathematics) you find a decomposition of the space of all curvature structures in terms of Lie and Jordan algebras. And you find how elegantly electrodynamics and gravitational waves fit into the curvature play, look at the basic work of Lichnerowics. As an ,algebraiker' I like to write the curvature structure in the following triple form, generalizing the concept of Lie triples (the book of Otmar Loos is a nice generalization of Lie theory):
[x,y,z]=R(x,y)z. This concept generalizes the notion of a Lie triple to that one of a curvature triple, where only the Jacobiidentity is missing, but a reference to the bilinear form <,> is added in such a way, that R(x,y) is an element of the pseudoorthogonal Lie algebra. Note that the complete work of Ricci, Einstein and Weyl can be summarized as a decomposition of the space of curvature structures of Levi type (for Lie algebras). All this shows, that we do not yet understand this curvature space completely. Especially the gravitational wave aspect needs clarification. Hannes Tilgner