Any orthonormal set extends to an orthonormal basis, over any field
of characteristic not 22. This is a special case of Witt's theorem.
EDIT: In response to Vipul's comment: The proof of Witt's theorem is constructive, and leads to the following recursive algorithm for extending an orthonormal set lbracev1,ldots,vrrbracelbracev1,ldots,vrrbrace to an orthonormal basis.
Let e1,ldots,ene1,ldots,en be the standard basis of KnKn, where eiei has 11 in the ioperatornamethioperatornameth coordinate and 00 elsewhere. It suffices to find a sequence of reflections defined over KK whose composition maps vivi to eiei for i=1,ldots,ri=1,ldots,r, since then the inverse sequence maps e1,ldots,ene1,ldots,en to an orthonormal basis extending v1,ldots,vrv1,ldots,vr. In fact, it suffices to find such a sequence mapping just v1v1 to e1e1, since after that we are reduced to an (n−1)(n−1)-dimensional problem in ep1erpep1erp, and can use recursion.
Case 1: q(v1−e1)ne0q(v1−e1)ne0, where qq is the quadratic form. Then reflection in the hyperplane (v1−e1)perp(v1−e1)perp maps v1v1 to e1e1.
Case 2: q(v1+e1)ne0q(v1+e1)ne0. Then reflection in (v1+e1)perp(v1+e1)perp maps v1v1 to −e1−e1, so follow this with reflection in the coordinate hyperplane ep1erpep1erp.
Case 3: q(v1−e1)=q(v1+e1)=0q(v1−e1)=q(v1+e1)=0. Summing yields 0=2q(v1)+2q(e1)=2+2=40=2q(v1)+2q(e1)=2+2=4, a contradiction, so this case does not actually arise.
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