Even-dimensional Clifford spinor dimension (${\dim }_{\mathbb{C}}{S}_D=2^{D/2},{\dim }_{\mathbb{C}}{S}_D^{\pm }=2^{D/2-1}$)

After complexification, pair the $D=2n$ Clifford algebra generators into $n$ fermionic creation and annihilation operators. Their occupation basis has $2^n$ states, giving the irreducible complex Dirac module. The chirality operator separates its even and odd occupation subspaces, each with $2^{n-1}$ states. Majorana reality conditions concern real dimensions and depend on spacetime signature; they must not be imposed in every even dimension indiscriminately.