A quick geometrical proof that \(G_ 2(K)\) is maximal in \(P\Omega _ 7(K)\) (Q1103193)

On a nonsingular quadric Q in a projective space PG(6,K) with totally singular planes on it some configuration \({\mathcal F}\) of planes, lines and points is given. As \textit{J. Tits} showed [Inst. haut. Étud. sci., Publ. Mat. No.2, 13-60 (1959; Zbl 0088.372)] the stabilizer of it in \(P\Omega_ 7(K)\) is the Chevally exceptional group \(G_ 2(K)\). By considering the group of projectivities of Q the maximality of \(G_ 2(K)\) as a subgroup in \(P\Omega\) (K) is shown. Furthermore the author gives some remarks on more algebraic proofs of the maximality.