Lemma 3 in complex coordinates: isometries preserving $\{P(z,\bar z)=0\}$ multiply $P(z,\bar z)$ by $\pm1$

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Lemma 3 in complex coordinates: isometries preserving $\{P(z,\bar z)=0\}$ multiply $P(z,\bar z)$ by $\pm1$ · Prove2Me