$V$ need not decompose into $W$ and $W^{\perp}$

Question

Let $B$ be a nondegenerate symmetric bilinear form on a vector space $V$ and let $W \subset V$. I want to show that $V$ need not decompose as a direct sum of $W$ and $W^{\perp}$.

I know that if I can find a $B$ such that $B$ is indefinite then I will be able to find a vector which is in both $W$ and $W^{\perp}$. However, I'm struggling to find such a $B$.

Answer 1

Let $B=\begin{bmatrix} 1 \\ & -1 \end{bmatrix}$. So clearly, $B$ is symmetric and non-degenerate. By $e = (1,1)^T$ is orthogonal to itself.