The Duffin Kemmer Petiau relativistic wave equation exists in en obscure corner of the history of QFT. It was used in the XX century to describe the properties of spin zero and spin one mesons. At the heart of the DKP theory is the meson algebra

$$\beta_\rho \beta_\mu \beta_\nu + \beta_\nu \beta_\mu \beta_\rho = g_{\mu \nu} \beta_\rho + g_{\mu \rho} \beta_\nu,$$ which has nontrivial 5 and 10 dimensional representations. This algebra permits the construction of a Dirac-like first order wave equation

$$ (i \beta_\mu \partial^\mu - m) \phi = 0.$$

From the point of view of 3+1 Poincaré covariant quantum field theory, this is a puzzling construction, since the propagating degrees of freedom comprise a mixture from different irreps of the Lorentz algebra. In particular, the spin one field is built from both chiral and non-chiral representations.

Now, the 4+1 Poincaré algebra has rank 3 and therefore its irreducible representations are indexed by three Casimir operators instead of two. The new Casimir is lineal in the momenta:

$$\mathcal{C}3=\frac{1}{2^3}\epsilon^{\alpha\beta\mu\nu\rho} M{\alpha\beta} M_{\mu\nu} P_\rho.$$

In this talk we show that the eigenvalue equation for this third extradimensional Casimir produces several linear equations in unconventional representations, including the DKP spin zero and spin one constructions. In this work we report the construction of a covariant basis for the Jordan-Lie algebra of operators acting on the 5 and 10 representations of the 4+1 Lorentz algebra, and the classification of kinematical operators and therefore of relativistic wave equations for these fields. As an application, we show how the 5 and 10 dimensional meson algebras arise as the restriction to 3+1 dimensions of these Jordan-Lie algebras.