= Criterion for membership in a diagonal basis
A vector $v$ can be included in a diagonal <basis> for a real <quadratic form> precisely when $v\ne0$ and either its quadratic value is nonzero or it lies in the <radical of a bilinear form>. A nonzero quadratic value gives the direct splitting $\mathbb Rv\oplus v^\perp$ and induction. A nonzero radical vector can be included in a radical basis. Conversely, an <isotropic vector> in a diagonal basis is orthogonal to every basis vector, and must lie in the radical.
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