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Because of the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern-0.1em \ell}_{2,3}(\Bbb{R})$, it is possible to complexify the spacetimeClifford algebra ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})$ by adding oneadditional timelike dimension to the Minkowski spacetime. In a recent work weshowed how this treatment provide a particular interpretation of Diracparticles and antiparticles in terms of the new temporal dimension. In thisarticle we thoroughly study the structure of the real Clifford algebra ${C\kern -0.1em \ell}_{2,3}(\Bbb{R})$ paying special attention to the isomorphism${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$and the embedding ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})\subseteq{C \kern -0.1em\ell}_{2,3}(\Bbb{R})$. On the first half of this article we analyze the Pin andSpin groups and construct an injective mapping$\operatorname{Pin}(1,3)\hookrightarrow\operatorname{Spin}(2,3)$, obtaining inparticular elements in $\operatorname{Spin}(2,3)$ that represent parity andtime reversal. On the second half of this paper we study the spinor space ofthe algebra and prove that the usual structure of complex spinors in ${C \kern-0.1em \ell}_{1,3}(\Bbb{C})$ is reproduced by the Clifford conjugation innerproduct for real spinors in ${C \kern -0.1em \ell}_{2,3}(\Bbb{R})$.

Complexifying the spacetime algebra by means of an extra timelike dimension: Pin, Spin and algebraic spinors