= Symmetric bilinear diagonal-parallel lemma
{title2=$S(u,v)=V(u)v+V(v)u$}
Let $S:W\times W\to W$ be a symmetric <bilinear map> on a finite-dimensional real <vector space>. If $S(v,v)$ is parallel to $v$ for every $v$, then it has the displayed form for one <covector> $V$. Write $S(e_i,e_i)=\alpha_i e_i$ in a <basis>. Applying the condition to $e_i+e_j$ and $e_i-e_j$ excludes every component outside their span and gives $S(e_i,e_j)=(\alpha_j e_i+\alpha_i e_j)/2$. Set $V(e_i)=\alpha_i/2$ and extend by <bilinearity>. Dimension one follows by the same definition. The <trace> identity $S^a{}_{ac}=(n+1)V_c$ proves uniqueness without depending on the chosen <basis>.
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