Geodesic
On the Index of a Quadratic Form
Canadian mathematical bulletin, Tome 1 (1958) no. 3, p. 180

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Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one. 
Wild, Jonathan. On the Index of a Quadratic Form. Canadian mathematical bulletin, Tome 1 (1958) no. 3, p. 180. doi: 10.4153/CMB-1958-019-8
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