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Subplanes of finite projective planes
Lemma. Let be a subplane of a projective plane . Let (o,e,u,v) be a quadrangle contained in the given subplane and let (K,T) be the corresponding ternary field. Then is a subfield of (K,T). The subplane is a Baer subplane of if and only if the following three conditions hold.
Lemma. Let V be a three-dimensional vector space over a skewfield K. Let be a subplane of PG(V). Then there exist a subskewfield P of K and a basis e1,e2,e3 of V such that The subplane is a Baer subplane if and only if K is of dimension 2 over P as a left vector space and as a right vector space, respectively.