For a paper, I am looking for a reference in the literature for the following theorem:
Given a symmetric non degenerate bilinear form $\phi$ on a finite dimensional vector space , with $\dim >1$, and $\phi(e_i, e_j)=0$ if $i+j>n+1$, there is a basis of $_1$,...,$_$ of V such that the matrix associated to $\phi$ with respect to $_1$,...,$_$ is the anti-diagonal matrix.
A proof for this statement is given at
Anti-diagonal matrix symmetric bilinear form
(though the original poster left out an important condition)
Thank you in advance