Let $X$ be a real finite dimensional linear space. Let $B:X\times X \rightarrow \mathbf{R}$ be a bilinear symmetric non-degenerated form. Let $M$, $N$ be totally isotropic subspaces with the same dimension $r$ such that $M\cap N=\{0\}$.
Assume that $e_1, ...,e_s$ are linearly indedendent in $M$, $f_1,...,f_s$ are linearly independent in $N$ (where $s< r$) and $B(e_i,f_j)=\delta_{i,j}$ for $i,j=1,...,s$.
Is it possible to extend $e_1,...,e_s$ to a basis $e_1,...,e_r$ in $M$, $f_1,...,f_s$ to a basis $f_1,...,f_r$ in $N$ in such a way that $B(e_i,f_j)=\delta_{i,j}$ for $i,j=1,...,r$?
I know only that for each basis $e_1,...,e_r$ in $M$ there exists a basis $f_1,...,f_r$ in $N$ such that $B(e_i,f_j)=\delta_{i,j}$ for $i,j=1,...,r$ (Bourbaki, Algebra, Chap.11, $\S$4,2, Prop.2)
Thanks.