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So, in contrast, must a spinor parameterized axial vector (pseudovector) [itex]\bar\psi\gamma^\mu\gamma\psi[/itex] be spacelike? Why is so?

P.S. [itex]\psi[/itex] denotes the components of a Dirac spinor expressed in terms of a column matrix, [itex]\bar{\psi}=\psi^\dagger\gamma^0[/itex] is the adjoint spinor to [itex]\psi[/itex], [itex]\gamma^\mu[/itex] denote the Dirac matrices, and [itex]\gamma=\gamma^0\gamma^1\gamma^2\gamma^3[/itex].

In addition, please help me confirm the following. Does the notion that under spatial inversion the components of a polar vector change a sign while the components of an axial vector (a pseudovector) keep invariant only apply to Riemannian geometry? Because it's like in pseudo-Riemannian geometry, such as Minkowski space, under spatial inversion the spatial components of a vector change a sign while the temporal component of it keeps invariant.