Suppose $(V,f)$ is a nonsingular orthogonal geometry with dimension $n$, and $W$ is a totally isotropic space with dimension $r$. How to find such a totally isotropic space $N$ with dimension $r$ such that
1)$V=W^\bot \oplus N$
2)$V=W \oplus H\oplus N$ where $H=W^\bot \cap N^\bot$
In addition, if the first condition holds, then the second one is true.
Suppose there's a base $\{a_1,\dots,a_s,a_{-1},\dots,a_{-(n-s)}\}$ that $f$ under this base is $\begin{pmatrix} I_s& 0\\0&-I_{n-s}\end{pmatrix}$. And suppose $W=\langle a_1+a_{-1},...,a_r+a_{-r}\rangle$. Then how to construct such a $N$? It seems that $N=\langle a_1+a_{-(n-2r+1)}, \dots ,a_r+a_{-(n-r)}\rangle $ satisfies the conditions, but I not pretty sure. Am I wrong or right? If I'm right, then how to prove it? Is there an clearer way to see this? Thanks in advance.
$\textbf{EDIT}$:After reading some suggestions on meta, I realized that I was not explicit with my post and efforts. As I've mentioned above, I $\textbf{seemed}$ to find an $N$ that satisfied the conditions. But when I examined this $N$ in the general case $n$, I found that the calculation was too difficult. So I worked with some simple cases when $n=5$, $r=s=2$ and $n=9$, $r=s=3$. In these cases, the first condition is always satisfied. But as for the second one, it's not clear because the basis of $H$ is really complicated.
So I'm thinking that my answer might be wrong or I chose a "complicated" basis. Sorry that this might be a silly question. Thanks again.