Pseudo-unitarity of the spinor Lorentz representation ($S^{†}{\gamma }^{0}S={\gamma }^0$)

The spinor Lorentz matrices preserve the indefinite form defined by ${\gamma }^0$: $S^{†}{\gamma }^{0}S={\gamma }^0$. This follows from the gamma-matrix adjoint identities for infinitesimal generators and extends to finite connected transformations. Consequently the Dirac adjoint transforms with $S^{-1}$, even though a boost matrix is not ordinarily unitary.