If $\beta$ is a symmetric bilinear form, non-degenerate, on a $K$-vector space $V$, then on any subspace $W$ of $V$, $\beta : W \times W \rightarrow K $ is also a symmetric bilinear form non-degenerate .
Do you have any clue if this statement is right? I found a counter-example if $\beta : W \times W' \rightarrow K $, but is it true if you take twice the same subspace?