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The Cayley-Klein (CK) formalism is applied to the real algebra ${so}(5)$ bymaking use of four graded contraction parameters describing in a unifiedsetting 81 Lie algebras, which cover the (anti-)de Sitter, Poincar\'e,Newtonian and Carrollian algebras. Starting with the Drinfel'd-Jimbo real Liebialgebra for ${so}(5)$ together with its Drinfel'd double structure, we obtainthe corresponding CK bialgebra and the CK $r$-matrix coming from a Drinfel'ddouble. As a novelty, we construct the (first-order) noncommutative CK spacesof points, lines, 2-planes and 3-hyperplanes, studying their structuralproperties. By requiring to deal with real structures, it comes out that thereexist 63 specific real Lie bialgebras together with their sets of fournoncommutative spaces. Furthermore, we find 14 classical $r$-matrices comingfrom Drinfel'd doubles, obtaining new results for the de Sitter ${so}(4,1)$ andanti-de Sitter ${so}(3,2)$ and for some of their contractions. These geometricresults are exhaustively applied onto the (3+1)D kinematical algebras, not onlyconsidering the usual (3+1)D spacetime but also the 6D space of lines. Weestablish different assignations between the geometrical CK generators and thekinematical ones which convey physical identifications for the CK contractionparameters in terms of the cosmological constant/curvature $\Lambda$ and speedof light $c$. We finally obtain four classes of kinematical $r$-matricestogether with their noncommutative spacetimes and spaces of lines, comprisingall $\kappa$-deformations as particular cases.

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications