The key point in writing an action for spinors is the existence of a Clifford algebra (expanded by the gamma matrices) $$\{\gamma^a,\gamma^b\} = 2\eta^{ab}\mathbf{1},$$ where the index $a$ runs from $0$ to $3$ (or $0$ to $D-1$, where $D$ is the spacetime dimension).

The whole basis for the algebra is given by the $\gamma$'s and all possible products... due to last equation, only antisymmetric products contribute.

It's possible to show that for every even dimensional spacetime the (antisymmetric) product of all $\gamma$'s, i.e., $\gamma^* \propto \gamma^0\cdots \gamma^{D-1}$ allows to define projectors $$P_+^2 = P_+,\quad P_-^2 = P_-, \quad P_+ P_- = 0.$$

These projectors serve to split the spinor into pieces, $$\psi_{\pm} = P_{\pm} \Psi,$$ with $\Psi$ the Dirac spinor, and $\psi_\pm$ the Weyl (or chiral) ones.

The key point in writing an action for spinors is the existence of a Clifford algebra (expanded by the gamma matrices) $$\{\gamma^a,\gamma^b\} = 2\eta^{ab}\mathbf{1},$$ where the index $a$ runs from $0$ to $3$ (or $0$ to $D-1$, where $D$ is the spacetime dimension).

The whole basis for the algebra is given by the $\gamma$'s and all possible products... due to last equation, only antisymmetric products contribute.

It's possible to show that for every even dimensional spacetime the (antisymmetric) product of all $\gamma$'s, i.e., $\gamma^* \propto \gamma^0\cdots \gamma^{D-1}$ allows to define non-trivial projectors $$P_+^2 = P_+,\quad P_-^2 = P_-, \quad P_+ P_- = 0.$$

These projectors serve to split the spinor into pieces, $$\psi_{\pm} = P_{\pm} \Psi,$$ with $\Psi$ the Dirac spinor, and $\psi_\pm$ the Weyl (or chiral) ones.

Conclusion

The fact that Dirac spinors can be split into Weyl ones is due to the dimensionality of spacetime. In odd dimension, the projectors are trivial because they are $0$ and $1$ respectively.