quantum field theory - How to derive the form of the parity operator acting on Lorentz spinors?

I'm reading Berestetskii (Volume 4 of Landau & Lifshitz) section 19 on inversion of spinors. Berestetskii says parity $P$ maps undotted spinors into dotted spinors and vice-versa as $\xi^{A}\rightarrow i\eta_{\dot{A}}$ and $\eta_{\dot{A}}\rightarrow i\xi^{A}$. I think he means there are parity tensors $P_{\dot{A}B}$ and $P^{A\dot{B}}$ which act as, $$ \eta_{\dot{A}}=P_{\dot{A}B}\xi^{B} \\ \xi^{A}=P^{A\dot{B}}\eta_{\dot{B}} $$ and which satisy $P^{2}=-1$, $$ P^{A\dot{C}}P_{\dot{C}B}=-\delta^{A}_{B} \ . $$ Berestetskii's map then says, $P_{\dot{A}B}=P^{A\dot{B}}=i\delta_{AB}$ which is only true in (presumably) the rest frame of the fermions. I've tried to derive the components of the parity tensors by imposing various mathematical and physically motivated restrictions, but I can only get $P^{A\dot{B}}$ diagonal with entries $a,1/a^{*}$ and $P_{\dot{A}B}$ diagonal with entries $-1/a,-a^{*}$. Berestetskii's parity is recovered by taking $a=i$ and so my question is, "What are the extra restrictions needed to be able to set $a=i$?"