the direction of the velocity $\vector{V}$, or momentum $\vector{P}$. The factor of $\basis{4}$ in the middle of eq. (2.5b) causes the spacetime split \[see eq. (2.4)\] of the four-momentum, where the fourth component $p^4$ is physically interpreted as the [*energy*]{} $E$ in eq. (2.5c).

We define the following algebra involutions, $${\cal T}(\Gamma)=-\basis{1}\basis{2}\basis{3}\ \Gamma
\ \basis{1}\basis{2}\basis{3}, \eqno(2.6a)$$ $${\cal P}(\Gamma)=-\basis{4}\ \Gamma\ \basis{4}, \eqno(2.6b)$$ which correspond geometrically to reflections. The first inverts timelike geometry, hence is called [Time Reversal]{} in physics. Basis elements which are invariant under this involution (e.g.$\basis{1}\basis{2}$) are called [*spacelike*]{}, while those which acquire a minus sign (e.g. $\basis{1}\basis{4}$) [*timelike*]{}. We will have more use for the second, the contrapositive inversion of space called the [Parity Transformation]{}[@Drell64], under which many (but not all) physical laws are invariant. The composition of the two is the [*main involution*]{} of the algebra, $${\cal PT}(\Gamma)={\cal TP}(\Gamma)=-\basis{1}\basis{2}\basis{3}\basis{4}
\ \Gamma\ \basis{1}\basis{2}\basis{3}\basis{4}, \eqno(2.6c)$$ inverting the odd geometry (i.e. vectors and trivectors).

The anti-involutions associated with eq. (2.6b) and (2.6c) are respectively the “dagger” and “bar” (equivalently the [*Hermitian*]{} and [*Dirac*]{} conjugates respectively). They are related, $$\overline{\Gamma}=-\basis{4}\ \Gamma^{\dagger} \ \basis{4}, \eqno(2.7a)$$ $$\overline{\bf e}_{\mu}=-\basis{\mu}, \eqno(2.7b)$$ $$\basis{4}^{\dagger}=-\basis{4},\ \ \ \basis{j}^{\dagger}=+\basis{j}
\ (j=1,2,3), \eqno(2.7c)$$ where the “bar” reverses the order of 